COMP 304Computer Graphics II

LECTURE 8

MOTION CONTROL – FORWARD KINEMATICSDr. Mehmet Gokturk

Asst. Prof., Gebze Institute of Technology

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Some Timeline The Illusion of Motion 1824, Peter Mark Roget,"Persistence of Vision with Regard to

Moving Objects“ a series of images shown in rapid sequence can appear to move

fluidly (i.e. a flip book or film projector)

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Timeline Movies (1895) age of movie camera and projector begins

– experimentors discover they can stop the crank and restart it again to obtain special effects

(1914) Gertie, Windsor McCay (newspaper cartoonist)– first popular animation

(1928) Steamboat Willie, Disney– an early cartoon w/ sound– cartoons seem plausible as entertainment

(1933) King Kong, Willis O’Brien

(1930’s & 40’s) Golden age of cartoons– Betty Boop, Popeye, Porky Pig, Daffy Duck, Bugs Bunny, Woody Woodpecker, Mighty

Mouse, Tom & Jerry

(1937) Snow White, Disney– animated feature film– cost is $1.5M

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Timeline Movies (cont) (1982) Tron, MAGI

– movie with a computer graphics premise

(1984) Last Starfighter– computer graphics was used interchangably with actual models of the spaceship

(1993) Jurassic Park– computer graphics is used to create living creatures that are meant to appear

realistic

(1995) Toy Story, Pixar– full-length feature film done entirely with 3D computer animation

(2000) CyberWorld 3D, IMAX– 3D IMAX full-length feature film including characters from popular 3D movies such

as ANTZ and The Simpsons’ Homer

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Conventional Animation Techniques Drawing on film Multiple drawings Rotoscoping (project film of real actors onto

drawing paper) Stop motion animation Acetate cels, multiple plane cells

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Conventional Animation Process Storyboard Key frames drawn

– Straight ahead vs. pose-to-pose Intermediate frames filled in

(inbetweening) Trial film is made

(called a pencil test) Pencil test frames transferred to cels

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Conventional Animation Process

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Role of the Computer In-betweening

– artistic example: Hunger, Peter Foldes 1974

Disney’s CAPS system– scanned artist drawings are read in– "cels" are colored online (broad color palette, exact color matching)– compositing is done online (background, 2D drawings, 3D animation)– 3D effects can be created with 2D drawings (e.g. Beauty and the Beast)– used in every film since Beauty & Beast

3D graphical worlds– can experiment more easily with actor position, camera position– can perform more complex camera moves– exchange labor to create drawings with labor to build and animate 3D

world

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3D Animation 3D animation is similar to stop

motion animation

King Kong (1932)

Flash Gordon (1972)

http://www.stopmotionanimation.com/

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3D Animation Stop motion animation

(Nightmare Before Christmas)

3D keyframing(Luxo Jr.)

Performance animation and motion capture (Donkey Kong Country)

Which must be done straight-ahead and which can be animated pose-topose?

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Keyframing Key frames mark important visual transitions (extremes of

action) Inbetweening is creation of intermediate frames between

the key frames Can easily be calculated by computer

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Temporal Sampling

Film recording takes samples of an image at fixed time intervals– 24 frames per second for film– 30 frames per second for video

human eye "sees" continuous motion

Sometimes, fewer keyframes are required to describe the motion, especially for “pencil tests” or rough choreography (e.g., Lost World)

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No internal energy source and move only when an external force acts on them.

Read for use when: – physical laws encoded – initial conditions specified

Pools of water, clothing, hair, leaves

Smooth Motion Passive Physics

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Clothing (Geri’s Game)

Water (Antz)

“Rigid” body physics (crashing space pods in Phantom Menace)

Geri’s Game, Pixar Animation Studios

Smooth Motion Passive Physics

See examples

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User specifies keyframes (start, end, middle) User specifies constraints (e.g. laws of physics) System searches for minimum energy motion to accomplish

goals

A. Witkin and M. Kass,“Spacetime Constraints”,SIGGRAPH ‘88.

Smooth Motion Active Physics

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Smooth Motion Active Physics and Simulation

Control an animated character as we would control a robot

behavior is simulated

a "control system" sends proper signals to the character’s "muscles" over time

Mark Raibert’s leg lab at MIT

http ://www.ai.mit.edu/projects/leglab/

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Noise Motion We generally don’t want motion to be too smooth The eye picks up symmetries and smooth curves and interprets them

as artificial or fake By adding noise, we can add texture to smooth motion

K. Perlin, “An Image Synthesizer”,Computer Graphics, 19(3), July 1985.

Perlin, Improv system(K. Perlin and A. Goldberg, SIGGRAPH ‘96).

Applets: http://www.mrl.nyu.edu/~perlin/

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Noise Motion Motion capture (natural noise!)

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Camera Path Following A simple type of animation everything remains static

except the camera (walk throughs or flybys).

The camera just as any other object as far as orientation and positioning is concerned.

The user needs to construct a path through space for the observer to follow along with orientation information.

Path = key frame positioning + interpolation of the inbetween frames.

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Camera Path Following ways to deal with the view direction (1) The center of interest can be held constant while

observer position is interpolated along a curve

View Direction Vector between the observer position (POS) and the center of interest (COI)

This is useful when the observer is flying over an environment inspecting a specific location such as a building.

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Camera Path Following ways to deal with the view direction (2) A path for the center of interest can be constructed,

say from a series of buildings in an environment.

Often, the animator will want the center of interest to stay focused on one building for a few frames before it goes to the next building.

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Camera Path Following ways to deal with the view direction (3) Alternatively, the center of interest can be controlled by

other points along the observer path.

For example, observer position for the next frame can be used to determine the view direction for the current frame.

Sometimes this is too jerky. Some nth frame beyond the current frame can be used to produce a smoother view direction.

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Camera Path Following ways to deal with the view direction (4)

The center of interest can also be attached to objects in the animation.

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Path following Have position and orientation interpolation for key

framing now

Combining them , get general motions of rigid objects– Add scaling, get stretching/squashing

Path following: – Have keys only for position– how to change orientation “naturally”

Same techniques for camera motion

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Orientation along a path It’s natural to change orientation as things move Example: looking while walking

– Look in the direction one walks Tedious to specify orientations along the way Want to get directly from the path

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Frenet frame (Moving frame) At each point on the curve:

– Get Tangent vector– Get vector in general curvature direction

• In the plane of tangent and curvature vector– Vector orthogonal to the two

Math:

Everything is normalized then

wuvsPsPusPw

);()();(

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Frenet frame Curvature can be zero along extended parts

– Example: straight line

Solution: interpolate boundary frames– Differ only by rotation around the straight line

Zero curvature at a point:– Possible flip (ex. camera flips upside down)

Discontinuities in curvature:– Sudden changes of object orientation

These effects are NOT tolerable

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Alternatives Tangent vector is ok for objects

– Poor choice for cameras

For cameras: Look at the “center of interest”– Fixed COI: Always look at particular point– Separate path for COI

• Can animate this point separately using extra key positions

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Alternatives COI (center of interest) travels in front of the camera

– COI(s)=P(s+ds)– At the end, along the final tangent vector

Choose several ds, average– Smoothes motion– Trade-off: jerky motion vs. too static view direction

“Up” vector:– In the plane of view vector and global UP vector– Extra offset from this direction– Full key framing

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Path Following

w = P’(u)

P’’(u)

u = P’(u) x P’’(u)

v = w x u

Frenet Frame

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Curvature continuity

=0

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Look ahead

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Define “look-at” vector

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Define “up” vector

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Kinematics X Dynamics

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Kinematics

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Kinematics

Keyframing requires that the user supply the key frames

For articulated figures, we need a way to define the key frames

There are two ways to pose an articulated character – forward and inverse kinematics

Kinematics is the study of motion without regard to the forces that cause the motion Kinematics

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Articulated ModelsArticulated models:

– rigid parts– connected by joints

They can be animated by specifying the joint angles (or other display parameters) as functions of time.

t1 t2

qi q ti ( )

t1 t2

See example animation clips

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Some robotics is required !

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Drawing Articulated Figures

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Static Figure Transformations

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Forward Kinematics Control

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Example: 2-Link Structure

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Forward Kinematics

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Forward Kinematics

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Forward Kinematics

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Forward Kinematics

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Forward Kinematics

Hierarchical model - joints and links

Joints - 1, 2, or 3 Degree of Freedom

Joints - rotational or prismatic

Links - displayable objects

Pose - setting parameters for all joint DoFs

Pose Vector - a complete set of joint parameters

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T

data

R

S

Forward Kinematics

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T1.1 T2.1

T1.2 T2.2

T0

R2.2R1.2

R2.1R1.1

User modifiesRotation(new “pose vector”)

Re-traverse treeTo get new “pose”

Transformations at the arcs

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Forward KinematicsDescribes the positions of the body parts as a function of the joint angles.

1 DOF: knee1 DOF: knee 2 DOF: wrist2 DOF: wrist 3 DOF: arm3 DOF: arm

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q

Degree of FreedomDOF

Joint Limits

Joint Representation

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Degrees of Freedom

Joint Limits

Multiple

qy

w

gimbal lock

Axis-anglequaternions

Joint Representation

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Note

User interface representation may not be the same used for internal representation and operations

qy

w

Joint Representation

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Drawing Articulated Figure

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Standard method of describing relationship of one DOF to next

Used extensively in robotics

Used in some early animation systems

Multiple DOF joints represented by zero-length parameters

Denavit and Hartenberg

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Skeleton Hierarchy

hips

r-thigh

r-calf

r-foot

left-leg ...

Each bone transformation described relative to the parent in the hierarchy:

, , , , ,h h h h h hx y z q f s

, ,t t tq f s

cq

,ffq f

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Forward Kinematics

vs

y

x

z

w =vsv( , )

ffq fTR( )cqTR( , , )

t t tq f sTR( , , ) ( , , )

h h h h h hx y z q f sT R

vsvs

Transformation matrix for a sensor/effecter vs is a matrix composition of all joint transformation between the sensor/effecter and the root of the hierarchy.

w hv = x , , , , , , , , , , , =h h h h h t t t c ff s sy z v vp

S S p

, , , , ,h h h h h hx y z q f s

, ,t t tq f s

cq

,ffq f

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Example (1) : Manipulator + 3 Revolute Joints

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Example (2) : (1) + Ball-and-Socket Joint

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• Four link deep appendage with a branch after the second link

• The first and third links have two degrees of freedom, the others have one.

• The first frame is defined by rotation angles ((0,0),0,[(30,0),-15],[(-30,0),15]

• The last frame is defined by rotation angles ((-30,60),-80,[(90,30),-135],[(-90,30),135]

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Summary