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Keyframing

Lecture 10 Slide 2 6.837 Fall 2003

Keyframing

Motion of objects is described as a function of time from a set of key object positions (keyframes).

• Keyframes are drawn by skilled animator.

• Computer generates in-betweens using interpolation.

( )s t

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Keyframing

Complex movements may require multiple keys.

Slide 4

Interpolating Positions

Given positions:

find curve such that

( , , ), 0, ,i i ix y t i n

( )( )

( )

x tt

y tC

( ) ii

i

xt

yC

0u0 0 0( , , )x y t

1 1 1( , , )x y t

2 2 2( , , )x y t

( )tC

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Lecture 10 Slide 5 6.837 Fall 2003

Linear Interpolation

Simple problem:

Any ti:

0 0 0( , , )x y t

1 1 1( , , )x y t

2 2 2( , , )x y t

010 1 0 1

1 0 1 0

2 11 2 1 2

2 1 2 1

, ,

( )

, ,

t tt tx x t t t

t t t tx t

t t t tx x t t t

t t t t

0 1=0 and =1t t 0 1( ) 1x t x t x t

Lecture 10 Slide 6 6.837 Fall 2003

Polynomial Interpolation

An n-degree polynomial can interpolate any n+1 points.

The n+1 coefficients of an n-degree polynomial can be obtained from the n+1 points.

0 0 0( , , )x y t

1 1 1( , , )x y t

2 2 2( , , )x y t

parabola

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Lecture 10 Slide 7

Spline Interpolation

For large number of points (N), many polynomials of small degree can be used. Polynomials of small degree are faster and easier to control.

x t

t t t

8-degree polynomial

spline spline vs. polynomial

Lecture 10

Spline Interpolation

A cubic polynomial between each pair of points:

4 parameters (degrees of freedom) for each spline segment.

Input parameters:

4 points

2 points + 2 directions

2 30 1 2 3( )x t c c t c t c t

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Lecture 10 Slide 12 6.837 Fall 2003

Interpolating Key Frames

• Interpolation is not fool proof.

• The splines may undershoot and cause interpenetration.

• The animator must solve these types of side-effects.

Keyframing

Keyframe animation can be applied to different parameters

Parameter to interpolate:

position,

orientation,

deformation,

lights,

camera,

opacity?

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Physics Simulation

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Physics Simulation

Particles

Rigid bodies

Deformable bodies

Fluid dynamics

Vehicle dynamics

Characters

Definitions

Kinematics: The study of motion without consideration of the underlying forces

Dynamics: Study of physical motion (or more abstractly, the study of change in physical systems)

Forward Dynamics: Computing motion resulting from applied forces

Inverse Dynamics: Computing forces required to generate desired motion

Mechanics, Statics, Kinetics

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Particles

Particula

p(t)

p(t+t)

zyx ppp ,,p

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Kinematics of Particles

Position : x

Velocity: v = dx/dt

Acceleration: a = dv/dt = d2x/dt2

dtvx

Aproximações númericas

Exato se v0 não varia em [0, t]

Exato se v(t) não varia em t

tvxtx 00)(

ttvttxtx )()()(

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Aceleração

Acceleration a=a0

Velocity

Position 00

2

0

00

xva2

1vx

vaav

ttdt

tdt

Aproximações númericas

Exato se a não varia em [0, t]

Exato se a(t) não varia em t

2)(2

1)()()( ttattvttxtx

2

002

1)( tatvxtx o

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Forces

Multiple forces can add up to a single total force:

Forces cause change in momentum (accelerations)

M/total

itotal

fa

ff

Gravity

Gravity near Earth’s surface is constant:

f=m.g (g = -9.8 m/s2)

Gravity for distant objects:

f=Gm1m2/r2 (G=6.673×10-11 m3/kg·s2)

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Spring-Damper

Spring: f = -kx

k=spring constant

x=distance from rest state

Damper: f=-cv

c=damping factor

v=velocity along spring axis

Spring-damper: f=-kx-cv

Aerodynamic Drag

Drag force: f=(1/2)ρaccdv2

ρ=fluid density

ac=cross sectional area

cd=coefficient of drag (geometric constant based on shape of object, usually between 0 and 1, but can be higher)

v=velocity of the object relative to velocity of the fluid

Note: for simple cases, (1/2)ρaccd is constant

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Friction

Static friction: f ≤ fnμs

Dynamic friction: f = fnμd

fn=normal force

μs=coefficient of static friction

μd=coefficient of dynamic friction

Force Fields

Generic force fields can be created that use arbitrary rules to define a force at some location: f=f(x)

Examples: vortex, attractors, turbulence, torus…

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Collisions: Impulse

Momentum: p = m.v

Impulse: J=Δp

An impulse is a finite change in momentum

Impulses are essentially large forces acting over a small time

Modified momentum update:

p=p0+fΔt+J p/m = p0/m+fΔt/m+J/m

v = v0+aΔt+J/m

[kg.m.s-1] or [N.s]

Particle Simulation 1

Particle struct {

Vector3d Velocity;

Vector3d Position;

float Mass;

}

UpdateParticle(float dtime) {

Force = ComputeTotalForce();

Acel = Force/Mass;

Velocity = Velocity + Acel * dtime;

Position = Position + Velocity * dtime;

}

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Particle Simulation 2

Particle struct {

Vector3d Momentum;

Vector3d Position;

float Mass;

}

UpdateParticle(float dtime) {

Impulse = ComputeTotalImpulse(dtime);

Momentum = Momentum + Impulse;

Velocity = Momentum / Mass;

Position = Position + Velocity * dtime;

}

Integration

Explicit Euler method:

v=v0+aΔt

x=x0+vΔt

Other methods: Implicit Euler

Runge-Kutta

Crank-Nicholson

Multipoint

Leapfrog

DuFort-Frankel

Adams, Adams-Moulton, Adams-Bashforth

Optimize for:

• Stability

• Accuracy

• Convergence

• Performance

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Rigid Bodies

Angular Speed

lim

0 t

d

t dt

• The average angular speed, ωavg, of a rotating rigid object is the ratio of the angular displacement to the time interval

• The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero

• SI unit: radian per second (rad/s)

• Angular speed positive if rotating in counterclockwise

• Angular speed will be negative if rotating in clockwise

f iavg

f it t t

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Instantaneous Angular Acceleration

lim

0 t

d

t dt

• The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0

• SI Units of angular acceleration: rad/s²

Torque

• Torque, t , is tendency of a force to rotate object about some axis

•

• Torque is vector: Direction determined by axis of twist

t Fd F is the force

d is the lever arm (or moment arm)

Units are Newton m

Φ is the angle between F and r t Fr sin

t Fd

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Offset Forces

Torque resulting from offset force: τ=r×f

Total force:

Total torque:

iff res

)fr( ii rest

Rotational Inertia

• Rotational inertia or moment of inertia I is the resistance of an object to changes in its rotational motion.

• Rotational inertia I is the rotational equal of mass M.

• Rotational inertia depends on:

• Mass

• Distribution of that mass around the axis of rotation.

Units: kg·m2

I = ⌠

⌡ r2 dm

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Moments of Inertia

Rotational Inertia

zzzyzx

yzyyyx

xzxyxx

III

III

III

I

dmxy

dmzy

xy

xx

)(I

)(I 22

zz

yy

xx

I00

0I0

00I

I0

TAIAI 0

For arbitrary axis:

A=3x3 orientation matrix

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Torque and Rotational Inertia

• Torque T is the application of a force which changes the angular acceleration of the object.

Newton’s Second Law for Rotation:

IT

L I

L mvr mr2

Angular Momentum

Analogy between L and p

Angular Momentum Linear momentum

L = Iw p = mv

t = dL/dt

t = I .

F = dp/dt

F = m.a

Conserved if no net

outside torques

Conserved if no net outside

forces

Rigid body

Point particle

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Angular Momentum

L=Iω = AI0ATω

L=angular momentum

I=rotational inertia

ω=angular velocity

A=3x3 orientation matrix

Rigid Body Simulation

RigidBody struct { Vector Position; Vector Velocity; Vector Orientation; Vector AngVelocity; float Mass; Matrix RotationInertia; } UpdateRigidBody(float dtime) {

Force=ComputeTotalForce(); Torque=ComputeTotalTorque();

Aceleration = Force / Mass; Velocity = Velocity + Aceleration * dtime; Position = Position + Velocity * dtime;

AngAceleration = Torque * Inverse(RotationalInertia); AngVelocity = AngVelocity + AngAceleration * dtime; Orientation = Orientation + AngVelocity * dtime; }

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http://bulletphysics.org/

• Physics Engine

– particle

– rigid body

– soft body

• Created by Erwin Coumans (previously of Sony)

• Professional-grade library

• Open Source

• Free for commercial use

Bullet3D

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• Continuous & Discrete Collision

• Ray & Convex Sweep Test

• Flexible Algorithm Selection System. Ex:

– Broad Phase: Sweep & Prune

– Narrow Phase: GJK

Physics Features

• Concave & Convex Meshes

• All basic Primitives

Collision Shapes

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• Rigid body dynamics constraint solver

• Vehicle dynamics

• Character controller & solver

• Ragdoll physics

• Soft body dynamics

– cloth

– rope

– deformable volumes

Advanced Features

• Blender3D

• Maya

• And support for multiple formats:

– Collada (.dae)

– Wavefront (.obj)

Integrated with various tools

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Bullet3D Components

• Perform collision detection

• Resolve collisions & other constraints

• Provide objects' updated world transform

Physics Engine Obligations

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• #include btBulletDynamicsCommon.h

• include Bullet/src path (-L …)

• build using these libs:

– BulletDynamics

– BulletCollision

– LinearMath

Bullet Setup

• Create a Bullet World

• Create a Rigid Body

• Create a Collision Object

• Update the simulation each frame

• Use object's world transform to render

Default Use of Bullet3D

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• Look at Hello World Demo

• Create:

– btDiscreteDynamicsWorld

– btcollisionShape

– btMotionState

– btRigidBody

• Each frame:

– call stepSimulation on dynamic world

– use world transform to render your object

Integrating Bullet in your App

• btDiscreteDynamicsWorld

OR

• BtSoftRigidDynamicsWorld

• Both subclasses of:

– btDynamicsWorld

• What does a world do?

– manages your physics objects & constraints

– implements the update of all objects each frame

Create a Bullet World

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• Add it to the btDynamicsWorld

Create a Rigid Body

• btCollisionObject

• Provide:

– mass (0 if static)

– collision shape

– material properties

• frictional coefficient

• coefficient of restitution

Create a Collision Object

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• Plane

• Box

• Sphere

• Cone

• Convex Hull

• Triangle Mesh

Collision Shape Options

Selecting the Best Bullet3D Collision Shape

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• Call stepSimulation

– will do frame's physics

– updates the world transform for active objects

• via their btMotionState's setWorldTransform

– uses internal fixed timestep of 60Hz

– performs interpolation for variation

Update the simulation each frame

• World Transform contains:

– position

– orientation

Use Object's World Transform to Render

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• btScalar: - float

– define BT_USE_DOUBLE_PRECISION to use as double

• btVector3

– x,y,z,w

– for 3D positions and vectors

• btQuaternion & btMatrix3x3

– for 3D orientations & rotations

Basic Bullet Data Types

• btCollisionObject

– a world transform

– a collision shape

• btCollisionShape

– box, sphere, convex hull, triangle mesh

– a single collision shape can be shared among multiple

collision objects

• btCollisionWorld

– stores all collision objects

Key Collision Data Structures

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• Minimum Separation

– creates small gap

• Bullet3D uses 0.04

– 4 cm

– can be changed, but don't set to 0

Collision Margin