Readings Grad Report

1

Dynamics I

Dynamics Simulation

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2

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Assignments

Assignment 2 -- Keyframing Due Friday, Jan 12. Questions?

Assignment 3 -- Billiards To be given Wednesday.

Plan Physics 101 for rigid body animation

Today: translation and rotational dynamics Wed: Forces, impacts, and collisions Next Monday: Numerical integration.

But first…

Notes on Assignment #1 Time is on my side…

At each frame, object motion is usually based on“clock time” or time of simulation.

In real time systems… System controls the frame update.

At update…must know “clock time” of simulation. Time must be accurate to milliseconds. Motion Control Algorithms usually need the time or time

since last update as parameter.

Notes on Assignment #1 Getting time:

Don’t use time().

Do use: ftime(), clock()

GetSystemTime() -- Windows gettimeofday() -- UNIX.

Questions?

Motivational Films

Educational animations by Jim Blinn

JPL

Cal Tech

Now at Microsoft

3

Motivational Film

Mathematica: The Theorem ofPythagoras (1988)

Motivational Film

The Mechanical Universe (1984)

Let’s get started Physics for Rigid Body Dynamics

Today -- first half: Linear Motion Today -- second half: Rotational Motion Wednesday: Forces / Collisions Next Monday: Numerical Integration / Constraints

Good Notes

1997 SIGGRAPH Course Notes -- David Baraff(CMU) http://www.cs.cmu.edu/~baraff/sigcourse

Gamasutra Article http://www.gamasutra.com/features/19980703/quaternions

_01.htm

Let’s get started

an·i·ma·tion (n)a motion picture made from a series ofdrawings simulating motion by means ofslight progressive changes in the drawings

Rendering Pipelinecamera(captureslight)

syntheticimage

cameramodel

(focusessimulatedlighting)

processing

photoprocessing

tonereproduction

realscene

3Dmodels

Photography:

ComputerGraphics:

Photographicprint

4

An animation framework Create world Populate the world Initialize clock while not done do

Move / animate all objects Render all objects Generate frame Update clock

Object oriented approach

WorldmoveAllObjects ()renderAllObjects()

AniObjectgeometrymove ()render()

AniControl move ()

inhabits

Keyframe

Physics Simulation

Particle System InverseKinematics

Modeling Motion Three approaches

Hueristics Rules of thumb, guidelines, cheap hacks Keyframing, traditional cel animation

Simulation Use of physics Dynamics

Sampling Use of measured data MoCap

Kinematics vs. Dynamics Kinematic control

Refers to motion of object irrespective of theforces involved

E.g. Keyframing

Dynamic control Concerned with computing the underlying

forces which are then used to produce motion.

Using Physics in Animation To achieve physically accurate motion,

go to the source! Keyframing accurate physical motion is

tedious. Use dynamics

Use physics to calculate motion Removes control from animator!! Motion is physically realistic.

The Source: Sir Isaac Newton 1643-1727 Discovered

(amongst otherthings) Calculus Physics of Light Physics of Motion

Busy Man

5

Laws of Motion Law I

Every object in a state of uniform motion tends toremain in that state of motion unless an externalforce is applied to it. (Inertia)

Law II: The acceleration of a body is proportional to the

resulting force acting on the body, and thisacceleration is in the same direction as the force.

Law III: For every action there is an equal and opposite

reaction.

Terms For Linear Physical Motion

Mass Measure of the amount of matter in a body From Law II: Measure of the a body’s resistance to

motion

Velocity Change of motion with respect to time

Acceleration Change of velocity with respect to time

Force In short, force is what makes objects accelerate

Terms

Linear Physical Motion Momentum

Defined as mass x velocity Another way of stating Law I

Momentum is conserved Cannot change unless force is acted upon it

Terms

Mathematically defined Velocity Acceleration Force

Momentum

dtdsv /=

dtsddtdva //2

==

maF =

!

M = mv

Terms Said another way:

If we have force and mass, we can calculatemotion.

m

tFta

)()( =

!= dttatv )()(

!= dttvts )()( !!=2)()( dttats

!

M(t) = F(t)dt"

Physical units

You want physical motion, you need touse physical units

Newtonkg• m / sec2

pound (lb)s • ft / sec2

Force

m / sec2ft / sec2Acceleration

m / secft / secVelocity

Meter (m)foot (ft)Distance

kilogram (kg)slug (s)Mass

SI (Metric)EnglishQuantity

6

Physical units Is weight == mass?

Depends upon who you talk to

Our convention (used in Bourg book) Mass = amount of matter in a body Weight = Force resulting in acceleration due to

gravity. g = 9.8 m / sec2 or 32 ft / sec2

Unambiguous in SI units English units: slug = mass, pound = force

Vectors

Note that all quantities mentioned(except for mass) are vector quantities We are, after all, dealing in 3D motion

Center of Mass

Definition the location where all of the mass of the

system could be considered to be located. For homogenous solid bodies that have a

symmetrical shape, the center of mass is atthe center of body's symmetry, itsgeometrical center.

Initial value problems In order to animate an object, we need to

find s(t), for various values of t, given Initial position of object Initial velocity of object Initial acceleration of object Forces applied to object

All of the above are vector quantities

Projectile motion

Shoot a projectile out of a cannon Cannon is positioned at a given angle No further acceleration except for

gravity.

Projectile motion

analysis

-g

acceleration

7

Recall

m

tFta

)()( =

!= dttatv )()(

!= dttvts )()(

!!=2)()( dttats

Solving differential equations

Means: Analytically Numerically

Analytic solution For constant acceleration problem

Acceleration

Velocity

Position

0=xa ga

y!=

!cos0vv

x= gtvdtavv yy !=+= " ## sinsin

00

! +=+= tvxdtvxxx

)cos( 000 " 2

0002

1)sin( gttvydtvyy y !+=+= " #

Analytic solution We now have equations of motion:

tvxtx )cos()( 00 !+=

2

002

1)sin()( gttvyty !+= "

Given any time t, we can determine position

Analytic solution

Let’s see this in action Link

Dynamics

Parameters: Initial conditions (position, velocity,

accelleration) Forces applied (can be time varying)

Output: Position / Orientation at time t.

8

Numerical Integration Remember, integration required

We won’t always be fortunate enough to be ableto perform the integration using calculus

Must code different equations for differentmotions.

In animation, numerical integration is usuallyused Full discussion of Numerical Integration

techniques next week.

Quick and dirty hack Euler Integration

Leonard Euler 1707-1783 Studied:

Number theory Differential Equations Newtonian Physics Rotational Motion

A busy man

Numerical IntegrationObject properties

Position, orientationLinear and angular velocityLinear and angular momentummass

Forces at time t

Calculate accelerationsUsing mass, momenta

Update object properties

Euler Integration

aka Method of Finite Differences Approximates a curve with a series of

straight lines corresponding to curveduring a given Δt

Uses tangent on the curve at a givenpoint to guide the curve to the nextpoint.

Euler Integration

F (ti)

F ’ (ti)

F (ti+1) = F (ti) + F ‘ (ti)*Δt

Euler Integration

Equations x (t + Δt) = x(t) + (v(t) Δt) v (t + Δt) = v(t) + (a(t) Δt) M(t + Δt) = M(t) + (F(t) Δt)

Approximating x, v, and M at time t + Δtby using tangents at time t.

9

Euler Integration

Euler Integration is fast, intuitive, andeasy to code however, It is also a very crude approximation to the

integral The smaller the Δt, the better the solution.

Also, the more work that is required.

Putting it all together…so far Step 1

Calculate Forces, F(t)

Step 2 update position (integrate velocity)

s(t +Δt) = s(t) + v(t)Δt update Momentum (integrate force/acceleration)

M(t +Δt) = M(t) + F(t) Δt

Step 3 Calculate velocities

v(t +Δt) = M(t +Δt)/m

Go to step 1

break

Putting it all togetherObject properties


Forces at time t



Let’s Talk About Rotation

Figures taken from Watt & Policarpo, “3D Games: RealTime

Rendering and Software Technology”

Let’s talk about rotation Two coordinate systems to consider

World space Coordinate system of space where object resides Known as inertial frame We’ll need everything w.r.t this for rendering.

Object space Local coordinate system of the object Origin is the center of mass of the object Known as body frame

Let’s talk about rotation

10

Transformation matrix

{ {objecttiontransforma

44434241

34333231

24232221

14131211

world

11!!!!

"

#

$$$$

%

&

'

!!!!

"

#

$$$$

%

&

=

!!!!

"

#

$$$$

%

&

o

o

o

w

w

w

z

y

x

mmmm

mmmm

mmmm

mmmm

z

y

x

4444 34444 21

Translation s(t)

Rotation = R(t)

Rotation Matrix Reinterpreting the rotation matrix

Each column gives the world coordinates of an axis in thelocal coordinate system

!!!

"

#

$$$

%

&

=

zzyzxz

zyyyxy

zxyxxx

rrr

rrr

rrr

tR )(

x y z

Rotation Matrix Factoids about the rotation matrix

R is orthogonal Each column vector is of unit length Determinant = 1 Inverse is equal to it’s transpose

R-1 = RT

Al the above must hold

Real reason why interpolation of this matrix directly does notwork!

Calculating position Taking into account rotation, the position (in world

space) of a point rbody (in object space)

Which is a different way of sayingbodyrtRtstr )()()( +=

{ {objecttiontransforma

44434241

34333231

24232221

14131211

world

11!!!!

"

#

$$$$

%

&

'

!!!!

"

#

$$$$

%

&

=

!!!!

"

#

$$$$

%

&

o

o

o

w

w

w

z

y

x

mmmm

mmmm

mmmm

mmmm

z

y

x

4444 34444 21

Angular velocity Amount of rotation with respect to time

Measured w.r.t. the body frame Given by a 3D vector ω(t)

Direction of ω gives the axis of rotation Magnitude of ω, | ω | gives the amount of rotation

Angular velocity

11

Angular velocity To obtain velocity at a point rbody with respect to

point in object space

Note that this is independent of linear velocity

Link

bodybody rttr != )()( "&

Angular Velocity

Apply to each axis in object space Recall

!!!

"

#

$$$

%

&

=

zzyzxz

zyyyxy

zxyxxx

rrr

rrr

rrr

tR )(

x y z

Angular Velocity

Apply to each axis in object space

)()()()()()( tRt

r

r

r

t

r

r

r

t

r

r

r

ttR

zz

zy

zx

yz

yy

yx

xz

xy

xx

!=

"""

#

$

%%%

&

'

"""

#

$

%%%

&

'

(

"""

#

$

%%%

&

'

(

"""

#

$

%%%

&

'

(= ))))&

Complete velocity equation

For a point rbody in object space Position

Velocity

bodyrtRtstr )()()( +=

bodyrtRtstr )()()( &&& +=

bodyrtRttvtr )()()()( !+= "&

translation

rotation



Calculate forces



Putting it all together Step 1

Calculate Forces, F(t) angular acceleration ω(t)

Step 2 Integrate position/rotation

s(t +Δt) = s(t) + v(t)Δt R(t +Δt) = R(t) + (ω(t)*R(t)) Δt /* CAREFUL HERE */ r(t +Δt) = s(t +Δt) + rbodyR(t +Δt)

Update Momentum M(t +Δt) = M(t) + F(t) Δt



12

Note about angular velocity Step 1

Calculate Forces F(t) angular acceleration ω(t)






Note about angular velocity Precipitated the use of quaternions Can use quaternions directly If q(t) represents rotation as a quaternion

then

where

!

˙ q (t) =1

2"(t)q(t)

!

"(t) = 0,"[ ]

Note about angular velocity Step 1

Calculate Forces, F(t) angular acceleration ω(t)


s(t +Δt) = s(t) + v(t)Δt q(t +Δt) = q(t) + 0.5 (ω(t)q(t)) Δt /* normalize to avoid problems */ R(t +Δt) = quatToRot (q(t +Δt) ) r(t +Δt) = s(t +Δt) + rbodyR(t +Δt)




Note about angular velocity

Note that rotat ion m atricies can st ill beused effect ively w ith a better num ericalinteg rat ion technique.

Putting it all together

What is missing? How about rotational force / acceleration This is known a Torque

Rotational acceleration

Consider

Apply force

Will result inrotation

13

Rotational acceleration

Observations Rotation occurs around the center of mass

(COM) of an object Force applied to point at a distance from

COM will cause a rotation

Torque Force applied to an object that results in rotational

acceleration. Dependent upon the distance from the center of mass

Where τ = Torque x = center of mass r = point where force is applied F = force applied

Fxr !"= )(#

Torque Now force can be seen as two separate

components: Force (Linear) Torque (Rotational)

Torque

More about mass When considering rotation, we need to know 3 things

about the mass of an object Magnitude Center of mass Description of how the mass is distributed in the object

This final item is given by the Inertia Tensor

Inertia tensor

Symmetric 3x3 matrix

!!!

"

#

$$$

%

&

=

zzyzxz

yzyyxy

xzxyxx

body

III

III

III

I

14

Inertia tensor

Each element of the tensor is calculatedby integrating the density of the objectat points q=(x,y,z)! += dmzyI xx )( 22

! += dmzxI yy )( 22

! += dmyxI zz )( 22

!"= xydmIxy

!"= xzdmIxz

!"= yzdmI yz

Inertia Tensor

Like in the linear case, The inertia tensor can be interpreted as

the resistance to accelerationmaF =

m

Fa =

!" &I=

!"# 1$== I&

Formulas for Inertia Tensor

For symmetrical bodies, I contains onlynon-zero components on the diagonal

!!!

"

#

$$$

%

&

=

zz

yy

xx

body

I

I

I

I

00

00

00


For common symmetric objects: Sphere with radius r and mass m

Cylinder with radius r, height h, where zaxis is coincident with the long axis ofcylinder

2

5

2mrIII zzyyxx ===

)3

1(

4

1 22 hrmII yyxx +==2

2

1mrI

zz=


For common symmetric objects: Rectangular box of sides a, b, c coinciding

to x, y, and z axes respectively

)(3

1 22cbmI

xx+=

)(3

1 22camI yy +=

)(3

1 22bamI

zz+=

Inertia Tensor

Important: With respect to time, in object space, I is

constant

To convert to world space:

Furthermore

T

body tRItRtI )()()( =

Tbody tRItRtI )()()( 11 !!

=

15

Let’s recap

For linear motion Velocity Acceleration Momentum Force

)(tsv &=

)()( tstva &&& ==

MvmmaF && ===

mvM =

Let’s Recap

For rotational motion Angular Velocity Angular Acceleration Angular Momentum Torque

)(tr&=!

)(t!" &=

)()()()()( tLttIttI && === !"#

)()( ttIL !=



Calculate forces



Putting it all together State of object at any given time position

rotation (in world coords)

momentum

angular momentum!!!!

"

#

$$$$

%

&

=

)(

)(

)(

)(

)(

tL

tM

tR

ts

tS

Putting it all together

Derivative of object state

!!!!

"

#

$$$$

%

&

'=

!!!!

"

#

$$$$

%

&

=

)(

)(

)()(

)(

)(

)(

)(

)(

)(

t

tF

tRt

tv

tL

tM

tR

ts

tS

(

)

&

&

&

&

&


Calculate Forces, F(t), τ(t)



Update Momentum (integrate accelleration) M(t +Δt) = M(t) + F(t) Δt L(t +Δt) = L(t) + τ(t) Δt

16


Calculate Forces, F(t), τ(t) Step 2

Integrate position/rotation s(t +Δt) = s(t) + v(t)Δt q(t +Δt) = q(t) + 0.5 (ω(t)q(t)) Δt /* normalize to avoid problems */ R(t +Δt) = quatToRot (q(t +Δt) ) r(t +Δt) = s(t +Δt) + rbodyR(t +Δt)

Update Momentum (integrate accelleration) M(t +Δt) = M(t) + F(t) Δt L(t +Δt) = L(t) + τ(t) Δt

Putting It all together Step 3

Calculate velocities (for next step) v(t +Δt) = M(t +Δt)/m I-1(t +Δt) = R(t +Δt)I-1

body R(t +Δt)T

ω(t +Δt) = + I-1(t +Δt)L(t +Δt)

Go to step 1

Questions?

Next Time

Forces / Collisions / Impulses

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