Phun with Physics

CSE 872 Dr. Charles B. OwenAdvanced Computer Graphics1

Phun with Physics

• The basic ideas• Vector calculus• Mass, acceleration, position, …


Some defs…

• Kinematics – The status of an object

• Position, orientation, acceleration, speed• Describes the motion of objects without considering factors that

cause or affect the motion

• Dynamics– The effects of forces on the motion of objects.


The basics• Let p(t) be the position in time. – We’ll drop the (t) and just say p

• Other values:– v – velocity– a - acceleration

2

2

dt

pd

dt

dva

dt

dpv

Velocity is the derivative of position

Acceleration is the derivative of velocity


Vector calculus

• Really, p(t) is a triple, right?– Think of these equations as three equations, one for

each dimension– This is vector calculus

dt

dpdt

dpdt

dp

tv

tv

tv

z

y

x

z

y

x

)(

)(

)(


Newton’s law and Momentum

• We know this one…

• Momentum

maF

mvM

Note: Force is the derivative of momentum


What about real objects?

• Think of a real object as a bunch of points, each with a momentum– We can find a “center of mass” for the object and treat the

object as a point with the total mass

• We have:– Mass, position vector, acceleration vector, velocity vector


Example: Air Resistance

• The resistance of air is proportional to the velocity– F = -kv

• We know F = ma, so: ma = -kv

kvdt

dvm

kvma

So, how can we solve?


Euler’s method

tm

kvvv

tm

kvv

m

kv

dt

dv

kvdt

dvm

ii

1


What about orientation?• We’ll start in 2D…• Let be the orientation (angle)– Easy in 2D, not so easy in 3D we’ll be back

• is the angular velocity around center of gravity• is the angular acceleration• Then…

dt

ddt

d


What’s the velocity at a point?

cpp rvr

p

Chasles’ Theorem: Velocity at any point is the sum of linear and rotational components.

Radians are important, here

cpop rvv

Perpendicular to cpSame length!Perpendicularized radius vector


Angular momentum

• Angular momentum– At a point?– For the whole thing?– M=mv

• Momentum equals mass times velocity• What does this mean?


We want to know how much of the momentum is “around” the center

pcpcp MrL

rcp

cpr

pMAngular Momentumof a point around c


Torque – Angular force

pcpcp MrL

pcpcp Fr

Torque at a point


Total angular momentum and moment of inertia

I

dprm

dprmr

dprmr

dpvmr

dpMr

dpLL

cpp

cppcp

cppcp

ppcp

pcp

cpcT

2)(

cpp rvr

I is the moment of inertia for the object.

pcpcp MrL


Moment of inertia

dprmI cpp 2)(

What are the characteristics?

What does large vs. small mean?

How to we get this value?


Relation of torque and moment of inertia

Ia

Ia


Simple dynamics

• Calculate/define center of mass (CM) and moment of inertia (I)

• Set initial position, orientation, linear, and angular velocities

• Determine all forces on the body• Linear acceleration is sum of forces divided by mass• Angular acceleration is sum of torques divided by I• Numerically integrate to update position, orientation,

and velocities


Object collisions

• Imagine objects A and B colliding

B

A

Assume collision is point on plane

n


What happens?

• ???

B

A

n


Our pal Newton…

• Newton’s law of restitution for instantaneous collisions with no friction– is a coefficient of restitution

• 0 is total plastic condition, all energy absorbed• 1 is total elastic condition, all energy reflected

nvnv '

What’s the consequence of “no friction”?


Impulse felt by each object

• Newton’s third law: equal and opposite forces– Force on A is jn (n is normal, j is amount of force)– Force on B is –jn– So…

nm

jvv

nm

jvv

BBB

AAA

'

'

No rotation for now…

B

A

n

We’ll need to know j


Solving for j…

BA

BA

BAB

BA

A

BABA

mmnn

nvvj

nvvnnm

jnvnn

m

jnv

nvvnvv

11

))(1(

)(

)()''(

nm

jvv

nm

jvv

BBB

AAA

'

'Then plug j into…


What about rotation?

A

AAA

AAA

I

jnr

nm

jvv

'

'

B

B

A

A

BA

BA

Inr

Inr

mmnn

nvvj

22 )()(11

))(1(


How to determine the time of collision?• What do I mean?

• Ideas?

• Avoid “tunneling”: objects that move through each other in a time step


Euler’s method

tm

kvvv

tm

kvv

m

kv

dt

dv

kvdt

dvm

ii

1

)),(()()(

...)(')),((

)()),((

ttvfttvttv

buttvttvfm

tkvttvf

We can write…


The problem with Euler’s method

...!

...)(''!2

)(')()( 0

2

000

n

nn

t

x

n

ttx

tttxtxttx

Taylor series:

We’re throwing this away!

Error is O(t2)

)()(')()( 2000 tOttxtxttx

Half step size cuts error to ¼!


Improving accuracy

• Supposed we know the second derivative?

)()(''!2

)(')()( 30

2

000 tOtxt

ttxtxttx

But, we would like to achieve this error without computing the second derivative.

We can do this using the Midpoint Method

)(,)),((2

)()()( 3000000 tOtttxf

ttxfttxttx

)()(')()( 2000 tOttxtxttx


Midpoint Method

• 1. Compute an Euler step

• 2. Evaluate f at the midpoint

• 3. Take a step using the midpoint value.

),( txftx

2

,2midpoint

tt

xxfx

midpoint)()( xttxttx


Example

m

tkvttvf

)()),((

),( txftx

2

,2midpoint

ttxxfx

midpoint)()( xttxttx Step 1

Step 2

Step 3


Moving to 3D

• Some things remain the same– Position, velocity, acceleration

• Just make them 3D instead of 2D

• The killer: Orientation– “It’s possible to prove that no three-scaler

parameterization of 3D orientation exists that doesn’t ‘suck’, forsome suitably mathematically rigorous definition of ‘suck’.” Chris Hecker


Orientation options

• Quaternions– We’ll use later

• Matricies– We’ll use this here.


Variables

• x(t)– Spatial position in time (3D)– Center of mass at time t– Assume object has center of mass at (0,0,0)

• v(t)– Velocity in space

• q(t) – Orientation in time (quaternion)


Angular velocity

• (t)– Angular velocity at any point in time– This is a rotation rate times a rotation vector

n

Strange, huh?


Note

• Angular velocity is instantaneous– We’ll compute it later in equations, but we won’t keep it

around.– Angular velocity is not necessarily constant for a spinning

object!• Interesting! Can you visualize why?


Angular velocity and vectors

• Rate of change of a vector is:

• This is why they like that notation– Change is orthogonal to the normal and vector– Magnitude of change is vector length times magnitude of

)()()( trttr


Derivative of the rotation quaternion• Woa…

)(,0[)( tqttq

Normalize this sucker and we can take a Newton step.

Be sure to scale (t) by t.


Velocity of a point (think vertex)

• Not just v(t), must include rotation!!!

)())()](,0([)()](,0[)( 10 tvtqtptqttp i

)()()()( 10 txtqptqtp ii Position of point at time t:

Just a reminder:

Velocity of point at time t:

And the magic:

Angular part Linear part

)(,0[)( tqttq

)()](,0))[()()]((,0[

)()](,0))[()()()()]((,0[

)()](,0[)()()](,0[

)())()](,0([)()](,0[)(

1

110

110

10

tvttxtpt

tvttxtxtqptqt

tvttqptqt

tvtqtptqttp

i

i

i


Force and torque

• Fi(t)– Force on particle i at time t– Vector, of course

)())()(()( tFtxtpt iii

)()( tFtF i

)()()()()( tFtxtptt iii

Torque on point i:

Total Force:

Total Torque:


Linear Momentum

)()( tMvtP Momentum:

M

tPtv

)()(

)()( tFtP And, the derivative of momentum is force:

M

tFtv

)()(


Angular Momentum

• Angular momentum is preserved if no torque is applied.

• L(t) is the angular momentum

)()( ttL


Inertia Tensor

)''(''''

'')''(''

'''')''(

)(22

22

22

iyixiiyiziixizi

iziyiizixiixiyi

izixiiyixiiziyi

rrmrrmrrm

rrmrrmrrm

rrmrrmrrm

tI

All that work typing this sucker in and it’s pretty much useless. Orientation dependent!


Inertia Tensor for a base orientation

Tiii

Tiibody ppIppmI 0000

dxdydzyxzyxyzdxdydzzyxxzdxdydzzyx

yzdxdydzzyxdxdydzzxzyxxydxdydzzyx

xzdxdydzzyxxydxdydzzyxdxdydzzyzyx

Ibody

))(,,(),,(),,(

),,())(,,(),,(

),,(),,())(,,(

22

22

22

22

22

22

00

00

00

hw

dw

dh

Ibox


Base inertia tensor to current inertia tensor

Tbody tRItRtI )()()(

)()()( ttItL

Relation of angular momentum and inertia tensor:


Bringing it all together

• Current state:

)(

)(

)(

)(

)(

tL

tP

tq

tx

tY

Position

Orientation

Linear momentum

Angular momentum

)(

)(

)(,0[

)(

)(

)(

)(

)(

)(

t

tF

tqt

tv

tL

tP

tq

tx

dt

dtY

)()()( 1 tLtIt

Velocity (may itself be updated)

)()( tFtF i

)()()()()( tFtxtptt iii


3D Collisions

• There are now two possible types of collisions– Not just vertex to face


3D collisions

• Vertex to face

• Edge to edge


We need…

• 3D collision detection– We’ll do that later

• Determine when the collision occurred– Binary search for the time– Bisection technique


Velocity at a point

))()(()()()( txtpttvtp aaaaa

)()()( 1 tLtIt

Position (center of gravity)

Position of point

Velocity (linear)

Angular velocity


Relative velocity

))()(()( tptptnv barel

Normal to object b

Vertex to face: normal for face

Edge to edge: cross product of edge directions

Positive vrel means moving apart, ignore it

Negative vrel means interpenetrating, process it

What does zero vrel mean?


New velocities for a and b

aaa M

tjntvtv

)()()(

))()()(()()( 1 tjntrtItt aaaa

bbb M

tjntvtv

)()()(

))()()()((()()( 1 tjntxtptItt bbbbb

)()()( txtptr aaa

)()()( txtptr bbb Let: Vector from center

of mass to point

Velocity (linear) update

Mass

Normal

Angular velocity update

Inertia tensor inverse

Equivalent equations for b


And j…

bbbaaaMM

rel

rtnrItnrtnrItn

vj

ba

)(()())(()(

)1(1111


Contrast 2D/3D

aaa M

tjntvtv

)()()(

aaa M

tjntvtv

)()()(

a

aaa I

tjnrtt

)()()(

))()()(()()( 1 tjntrtItt aaaa

2D 3D


Questions?

• 2D had two different j’s– Linear velocity and angular velocity

• 3D gets by with only one.– Why is this?


Other issues

• Resting contact– Bodies in contact, but – < vrel <

– Must deal with motion transfer

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Phun with Physics