chap 7 physically based animation 1 animation (u) chap 7 physically based animation cs dept., nctu,...

Chap 7Physically Based Animation

1Animation (U) Chap 7 Physically Based AnimationAnimation (U) Chap 7 Physically Based Animation CS Dept., NCTU, J. H. Chuang

Physically Based Animation

Forces are typically used to maintain relationships among geometric elements Accuracy vs. physical realism

Animators is not necessarily concerned with being accurate but rather with believabibity

Some forces may not relate to physics at all They may model constraints that an animator may wish

to enforce on the motion

CS Dept., NCTU, J. H. Chuang2Animation (U) Chap 7 Physically Based AnimationAnimation (U) Chap 7 Physically Based Animation


When modeling motion using physics principles, we need to decide the level at which to model the process Procedure

Computationally less expensive, easier to program Lacks flexibility

Purely physics Computationally expensive More flexible



When physical models are used, animator is relieved of lower level specification of motions Needs to be concerned with specifying high-level

relationships or qualities of the motion


Outline

Basic physics Spring meshes Particle systems Rigid body simulation Enforcing soft and hard constraints

A Good reference: Read Witkin and Baraff’s SIGGRAPH’01 course notes:

Physics-based modeling at http://www.pixar.com/companyinfo/research/pbm2001/index.html


Basic physics

Newton’s laws of motions

CS Dept., NCTU, J. H. Chuang6

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Animation (U) Chap 7 Physically Based AnimationAnimation (U) Chap 7 Physically Based Animation

Basic physics


mv

vkf

vkf

LLkf

vv

sdd

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air assuch medium, aough thr

velocity resists damper, similar to :Viscosity

:length spring theof velocity the toalproportion

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( :force Spring )


Basic physics


I

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describing Imatrix 3x3 a n;orientatioin change to

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Spring meshes

Flexible objects Flexibility is modeled by a spring-mass-damper

simulation simulating the reaction of the body to external forces.

Each vertex: a point mass Assigned by animators Evenly distributed among object’s vertices

Each edge: a spring Rest length is original edge length

Spring constants Arbitrary, assigned uniformly throughout the object


Spring meshes

Spring-mass-damper simulation As external forces are applied to specific vertices

Vertices will be displaced relative to other vertices The displacement induces spring forces, which will impart

forces to the adjacent vertices and reactive forces back to initial vertices.

Force propagation one time step at a time


sdrestcurrents vkLLkf )(


Spring meshesA simple example


Instant force F applied to V2.Length of springs E12 and E23 changes.Spring forces impart restoringforces to V1 and V2, and V2 and V3

Since external force is wrongly assumed constant throughoutthe time step, a spring simulation may numerically explode (the motions get larger and larger).

A smaller time step, or smallerspring constant, or larger massescan be used; but slow down the simulation.

A common method: add dampers.Animation (U) Chap 7 Physically Based AnimationAnimation (U) Chap 7 Physically Based Animation

Spring meshesDampers


A damper introduces an additional force in the spring that works against the velocity of the spring length, thus helping to limit the speed at which a spring changes length. Help to control the change in length and keep it

within some range that does not allow the simulation to explode.

sdrestcurrents vkLLkf )(Animation (U) Chap 7 Physically Based AnimationAnimation (U) Chap 7 Physically Based Animation

Spring meshes


Modeling with only object edges with spring dampers can result in a model that has more than one stable configuration. Additional spring dampers can help to stabilize the

shape.

If a cube’s edges are modeled with spring dampers, during applications of extreme external force is applied, the cube can turn Inside out.


Spring meshesAngular spring dampers


angular spring resists deviation to the rest angle between faces and imparts a torque along the edge that attempts to restore the rest angle.


Spring meshesVirtual springs


Virtual springs introduce forces into the system that do not directly model physical elements. For example, Example on the last page Virtual springs with zero rest lengths can be used

to constrained one object to lie on the other Virtual springs with non-zero rest lengths can be

used to maintain separation between moving objects.


Particle systems

Particles have been used to animate many behaviors, such as gases, water, fire, rubber, clothes, flocking, hair.



Particle systemsAssumptions

Particles do not collide with other particles Particles do not cast shadow on each other,

except in an aggregate sense Particles only cast shadow on the rest of

environment Particles do not reflect light Particles often have a finite life span



Particle systemsSteps in a frame

Generate any new particles Each new particle is assigned attributes Any particles that have exceeded their life

span are terminated The remaining particles are animated and their

shading parameters changes according to the controlling process

The particles are rendered



Particle generation For each frame, particles are generated

according to a controlled stochastic process User-specified distribution centered at the desired

average number of particle per frame Can be function of time

factor. scale a is and 1.0, to1.0- from

number random a returns Rand() where

Rand() particle of #

r

rn



Particle attributes Particle properties:

mass position velocity force accumulator age, lifespan rendering properties (shape parameters, color,

transparency) Each of the attributes is initialized when the

particle is created.



Particle life span

At each new frame, each particle’s lifetime is decremented by oneWhen the attribute reaches zero, the particle is removed from the system.



Particle animation Each particle is animated throughout its life.

Includes position, velocity, and rendering attributes Position and velocity

Users consider forces and compute the resultant particle acceleration, update its velocity, and update the position.

Color and transparency can be a function of time, its own life span remaining, its height, and so on. Shape can be a function of its velocity



Particle animation Forces on particles

Forces can be unary, particle pair, or environmental. Unary forces

Gravity, viscous drag Particle pair forces

Can be represented by springs and dampers, if desired. Environmental forces

Arise from a particle’s relationship to the environment



Forces on particles

Particles respond to forces We represent this using differential equations

m

fx

2nd order ODEm

fv

vx

1st order ODEs

phase space



Unary Forces

Forces that only depend on one particle

Gravity Wind FieldsViscous Drag

fdrag

v

f = mg f = kvwindf = -kdv



n-ary Forces

Forces that depend on n particles Example: binary forces between two particles

- spring and damper

Springs

||)|(| 0

ba

babasa lk

xx

xxxxf

||||

)()(

ba

ba

ba

babadk

xx

xx

xx

xxvv

af

bf



n-ary Forces: Spring Force If particle is located farther than the rest position, the

spring force needs to pull it back

If the particle is located nearer than the rest position, the spring force needs to push it away

Combine two cases:

0 rxx ba

0 , klkfa

0 rxx ba

0 , klkfa

0 ,)( sbasa klrxxkf

ba

babas xx

xxrxxk

)(

bx

ax

ba

ba

xx

xxl



n-ary Forces: Damping Forces

According to the law of energy conservation, a particle system consists of only masses and springs keep bouncing from each other after external forces disappear

Damping/viscous drag force resist motion, making a particle system gradually come to rest in the absence of external forces



n-ary Forces: Damping Forces (cont.)

It is highly recommended that at least a small amount of damping is applied to each particle

Excessive damping, however, makes a particle appear that floating in molasses (energy dissipates out too quickly, not responsive)



n-ary Forces: Damper Force If two particles are departing, the damper force needs

to pull them back

If two particles are approaching, the damper force needs to push them away

Combine two cases:

0)( lvv ba

0 , klkfa

0 ,)( dbada kllvvkf

ba

ba

ba

babad xx

xx

xx

xxvvk

)()()(

0)( lvv ba

0 , klkfa

bvav

ba

ba

xx

xxl



Spatial Forces

Forces that depend on nearby particles within a local region

Gravity, Lennard-Jones and electric potentials

Spatial data structures can optimize computations



Particle rendering Several methods

Model each particle as a point light source Each particle is rendered to a small graphical primitive

(blob). Particles that map to the same pixels in the image are

additive - the color of a pixel is simply the sum of the color values of all the particles that map to it

Model each particle as a textured billboard Polygon facing the viewer, texture

Rendered as Metaballs in off-line rendering; isosurfaces computed from particle-metaballs make quite convincing liquids.



Particle renderingParticle



Particle rendering Billboard texture




From Mark Harris’s workhttp://www.markmark.net/clouds/


Particle renderingBillboard texture

An impostor replaces a cloud with a billboard textured with an image of the cloud from a certain viewpoint. Image is updated only when translation of the

viewpoint introduces enough error in the image Impostors can be reused for many frames.

By using impostors, we are able to render cloudy scenes of hundreds of clouds and hundreds of thousands of particles at very high frame rates.


Particle renderingBillboard texture


Figure 6: Impostor generation and translation errormetric.

Impostor generation and translation error metric



Particle rendering

VideoCloud rendering by Niniane Wanghttp://www.ofb.net/~niniane/clouds/



Particle renderingMetaballs and isosurface

From “Screen Space Fluid Rendering with Curvature Flow “http://industrialarithmetic.blogspot.com/2009/01/

our-paper-screen-space-fluid-rendering.htmlAnimation (U) Chap 7 Physically Based AnimationAnimation (U) Chap 7 Physically Based Animation


Collision Detection

Determine when a particle has collided an object

Particle has collided if and only if

N

0xN

0xN

0xN

Object’s surface



Collision Response

What should we do when a particle has collided?

The correct thing to do is rollback the simulation to the exact point of contact

Easier to just modify positions and velocities

nv

tv

vN

newv

After the collision:

tnnew vvv

coefficient of restitution



Contact Forces

When the particle is on the collision surface a contact force resists penetration

Contact forces do not resist leaving the surface

Simple friction can be modeled

ffNf )( c 0)( fN

0cf 0)( fN

tff vk )( fNf 0)( fN



Structure of Particle Systems Separate the data structures and integration

Particle Systemtimeparticles

Particle

Solver

derivativesstate

state/derivatives

statesend data as6n vectors

xvfm

xvfm

xvfm

xvfm



Structure of Particle SystemsImplementation

Solver Interface

xvfm

GetDim

Deriv Eval

Get/Set Statexv

vf/m

System Solver

6



Structure of Particle SystemsImplementation

Solver Interface

particlesGetDim

Deriv Eval

Get/Set Statex1

v1

v1

f1/m1

System Solver

6n

n time

x2

v2

v2

f2/m2

xn

vn

vn

fn/mn



Physics Engines for Dynamics Simulation

Open Dynamics Engine (ODE) Free software http://www.ode.org/ high performance library for simulating rigid body

dynamics. It is fully featured, stable, mature and platform

independent with an easy to use C/C++ API. It has advanced joint types and integrated collision

detection with friction. ODE is useful for simulating vehicles, objects in

virtual reality environments and virtual creatures.




Havok Commercial software “Reactor” in 3ds MAX State-of-the-art game physics solution, for use with

in-house game animation systems Havok Destruction™ is the cross-platform tool for

simulation of rigid body destruction.




NVIDIA PhysX A powerful physics engine which enables real-time

physics in leading edge PC and console games. Is widely adopted by over 150 games, is used by

more than 10,000 registered users. Designed specifically for hardware acceleration by

powerful processors with hundreds of cores. Combined with the tremendous parallel processing capability of the GPU, PhysX will provide an exponential increase in physics processing power




AGEIA PhysX accelerator The world’s first dedicated physics processor designed

to support extreme physical gaming interactions. Its highly parallel, interactive PhysX cores are

optimized for dynamic, large-scale, physics processing to accelerate physical motion and interaction at a scale and quality far beyond that of general purpose processors.


Rigid Body Simulation

Various forces to be simulated are modeled. When the forces are applied to objects, they

induce linear acceleration (based on object’s mass) angular acceleration (based on mass’s distribution)

These accelerations are integrated over a delta time step to get changes of object’s velocities, which in turns integrated over a delta time step to produce changes in position and orientation.



Various forces to be simulated are modeled. When the forces are applied to objects, they

induce linear acceleration (based on object’s mass) angular acceleration (based on mass’s distribution)

Read Witkin and Baraff’s SIGGRAPH’01 course notes: Physics-based modeling http://www.pixar.com/companyinfo/research/

pbm2001/index.html


Rigid Body SimulationUpdate cycle



Continuous process vs. discrete time simulation Assume acceleration is constant over the delta time

step These accelerations are integrated over a delta time

step to get changes of object’s velocities, which in turns integrated over a delta time step to produce changes in position and orientation.

How to update? Euler integration method Runge –Kutta method: Second-order in magnitude of

error termCS Dept., NCTU, J. H. Chuang55Animation (U) Chap 7 Physically Based AnimationAnimation (U) Chap 7 Physically Based Animation


Physics-based Simulation

A procedure that generates a sequence of the states of a system based on physics laws

Newtonian lawselastic forces

windgravityfriction

…

xi

xi+1

△x

xixi+1= xi+ △x



Differential Equations

Differential equation describes the relation between an unknown function and its derivatives

Solving a differential equation is to find a function that satisfies the relation

Numerical solution of differential equations is based on finite-dimensional approximation


),())(( tftfdt

dxx

x


Ordinary Differential Equations Ordinary differential equation (ODE)

All derivatives are with respect to single independent variable, usually representing time

Time derivative of the unknown function

Unknown function that evaluates the state given time

Known function

00 at time vector state:)( ttx*We’ll show that a higher ODE can be transformed into this 1st order system soon!



Higher-Order ODEs

Order of ODE determined by highest-order derivative of solution function appearing in ODE

Equations with higher derivatives can be transformed into equivalent first-order system



Higher-Order ODEs (cont.) Given k-th order ODE

Define Original ODE equivalent to first order system

),,',...,,( )2()1()(

tyyyyfdt

yd kkk

ytx )(1

')(2 ytx

")(3 ytx

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'

'

)1(

3

2

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2

1

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x

x

x

x

x

x

x

kk

k

k



Visualizing Solution of ODE

x1

x2),( tfdt

dx

xx

point movinga :)(tx

s velocities':),( xx tf



Vector Field

The complete set of all solutions to the differential equation defines a vector field over (x, t) plane.

x1

x2

Think of this vector field as the sea, and the velocity of current at different places and time is defined by f(x,t)

),( tf xx



Initial Value Problem

For a particular initial value, the solution is a curve. Given the starting point, follow the integral curve

)( find ,)( and ),( Given 00 tttf xxxxx


x1

x2

Release a ball at any starting point and let it drift following the current. The trajectory swept out by the ball is an integral curve.


Numerical Solution of ODEs Instead of true integral curve, numerical solution

follow a polygonal path Each leg is obtained by evaluating the derivative

at discrete time steps Bigger steps, bigger errors

x1

x2



Issue I: Inaccuracy

Error turns x(t) from a circle into the spiral of your choice! May jump to other circles

x1

x2



Issue II: Instability

May diverge!

x1

x2


to Neptune!


Euler’s Method

Simplest numerical solution method Bigger time steps, bigger errors

x1

x2

),()()( tfhtht xxx


)(2

)()()(2

000 xxxx hththt


Euler’s Method (cont.)

Solves ODE using one-term Taylor-series

O(h2): 2nd order accurate

),( tf xx

),()()( tfhtht xxx

)( 00 txx

nnn hxxx 1

)( find ,)( and ),( Given 00 tttf xxxxx



Euler’s Method (cont.)

Truncation error in s single step : O(h2) The error in a complete solution is an

accumulation of all the single step errors


)O( iserror totalSo

).O( oferror an acquires stepeach where

steps need we),( compute To

2

2

0

hh

T

h

T/hTt x

Rigid Body Simulation Euler integration method (Book)

Explicit Euler integration method


),('1

1

nnnn

nn

yxhfyy

hxx



Drawbacks of Euler’s Method

Too inaccurate to be used in practice Inefficiency

Need to use small time-steps to avoid divergence Example:

http://www.cse.uiuc.edu/iem/ode/eulrmthd/

Improvement using the midpoint method Slope at midpoint is used


http://www.cse.uiuc.edu/iem/ode/eulrmthd/


Runge-Kutta methodThe Midpoint Method

a. Compute an Euler step

b. Evaluate f at the midpoint

c. Take a step using the midpoint value

))(( 0tfh xx

2

))(()(

)2

)((

00

0

tfhtf

tffmid

xx

xx

midfhtht )()( 00 xxAnimation (U) Chap 7 Physically Based AnimationAnimation (U) Chap 7 Physically Based Animation

x1

x2

Assume f depends on t only indirectly through x


Runge-Kutta method The Midpoint Method (cont.)

Solves ODE using two-term Taylor-series

Approximating f by Taylor-series

)(3

)(2

)()()(3

0

2

000 xxxxx ht

hththt

)()(')('),( xxx

xxxx ffdt

dftf

dt

d

dt

d

)(' )()( 000 xxxxx fff

00 2)( xx h

f )(')(2

)())(2

( 00000 xxxxx ffh

ffh

f

)(2

where 0xx fh



Runge-Kutta method The Midpoint Method (cont.)

)(3

)(2

)()()(3

0

2

000 xxxxx ht

hththt

0000 2)())(

2( xxxx h

ffh

f

)())(

2(

2 0000

2

xxxx ffh

fhh

)()())(2

()()( 300000 hOff

hfhhft

xxxxx

)())(2

()( 3000 hOf

hhft xxx


Turning O(h2) to O(h3)

Runge-Kutta method The Midpoint Method (Book)

Runge –Kutta method Symmetric w.r.t. the time interval Second-order in magnitude of error term


method.Euler explicit using evaluated is

interval of middle at the derivative the: thatNote

)),(2

,2

( ''1

1

nnnnnn

nn

yxfh

yh

xhfyy

hxx



Runge-Kutta 4th Order Method

Using a weighted average of slopes obtained at four points

),( 00 tf x

)2

,2

( 01

0

ht

kf x

t

)2

,2

( 01

0

ht

kf x

),( 030 htkf xx

)( 0 ht x

)()22(6

1)()( 5

432100 hOkkkktht xx

),( 001 thfk x

)2

,2

( 01

02

ht

khfk x

)2

,2

( 02

03

ht

khfk x

),( 0304 htkhfk x


Runge-Kutta 4th Order Method (Book)

Fourth-order Runge –Kutta method


)(226

1

),(

)2

,2

(

)2

,2

(

),(

543211

3'

4

2'3

1'2

'1

1

hOkkkkyy

kyhxhfk

ky

hxhfk

ky

hxhfk

yxhfk

hxx

nn

nn

nn

nn

nn

nn



)(6336

),(

)2

,2

(

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,2

(

),(

543211

3'

4

2'3

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'1

1

hOkkkk

yy

kyhxhfk

ky

hxhfk

ky

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yxhfk

hxx

nn

nn

nn

nn

nn

nn



Adaptive Step Size

Ideally, we want to choose h as large as possible, but not so large as to cause big error or instability

We can vary h as we march forward in time


What is optimal h if we want to have an error of as much as ε, and the current error is e?

Adaptive Step Size for Euler’s Method

Animation (U) Chap 7 Physically Based AnimationAnimation (U) Chap 7 Physically Based Animation CS Dept., NCTU, J. H. Chuang81

ek

ekek

h

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khh'

he

222)(

.Let *


Take Home Message

Don’t use Euler’s method Inaccuracy Inefficiency (or unstable)

Do use adaptive step size

Read Witkin and Baraff’s SIGGRAPH’01 course notes: Physics-based modeling http://www.pixar.com/companyinfo/research/

pbm2001/index.html

x1

x2


Motion equation for a rigid body

To develop the equations of motion for a rigid body, we need to discuss Linear force vs. rotational force

Torque Linear momentum vs. angular momentum Inertia tensor Equations


Orientation and rotational movement angular velocity

Angular velocity w(t) Direction: Axis of rotation, Speed of orientation in revolutions per unit of time

For linear motion, x(t) and v(t) are related by

How are R(t) and w(t) related? We need to find how the derivative of a vector in a

rigid body is related to w(t)


)()( txdt

dtv

Orientation and rotational movement angular velocity of a particle

Angular velocity of a point


sin)()()(

)()()(

)( oflocity angular ve)(

space. in world represnted is )(

)()()(

trtwtr

trtwtr

tptw

tr

trtxtp


p



)()()(

can write we together, thisPutting .)(

magnitude has )( of change ousinstantane theand

, )( and both lar toperpendicu is )( of change

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)()()( ofDirection

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)()(

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))(()(

)()()(

)( toparallel is where

,)()(Let

trtwtr

tbtw

tbtwatw

tbatw

trtwtr

twa

tbatr

Orientation and rotational movement angular velocity of a rigid body

Angular velocity of a rigid body Position and orientation of a rigid body are

represented by x(t), and R(t) x(t), and R(t) are used to transform the body-space

description into world space



R(t) specifies a rotation of the body about the center of mass It is a matrix of 3 orthogonal column vectors At time t,

represents the x-axis of body space in world space

Each column vector represents a vector from the origin

A point Q on the rigid body with local coordinate q


)()()()( where),(

0

0

1

)( 3211 tRtRtRtRtRtR


At time t, the derivative of the first column of R(t) is the rate of change of this vector. So


)()()()()()()(

)()()()(

321

321

tRtwtRtwtRtwtR

tRtRtRtR

)())()(()()()()()()(

)()()(

tvtxtqtwtvqtRtxqtRtq

txqtRtq


ba

b

b

b

aa

aa

aa

abba

abba

abba

ba

tRtwtRtwtw

tRtw

tRtwtRtwtRtwtR

z

y

x

xy

xz

yz

yxyx

zxzx

zyzy

ii

*

**

*

321

0

0

0

:Note

).()()()(such that matrix theis )( where

)()(

)()()()()()()(


A point Q on the rigid body with body-space coordinate q. Then the world-space location of q(t) is the result of

first rotating q about the origin and then translating it:


)())()(()()())()(()(

)())()()(()()(

)()(

)()()(

:)( of velocity theand

)()()(

*

1*

tvtxtqtwtvtxtqtw

tvtxtqtRtRtw

tvqtR

txqtRtq

tq

txqtRtq


This separates the velocity of a point on a rigid body into two components:A linear component v(t)A angular component w x (q(t)-x(t))

Orientation and rotational movementCenter of mass

Integration of the differential mass times its global position in the object


M

tqmtx

mM

ii

i

)()(


Orientation and rotational movementForces and torque

Linear force vs. rotational force


)())()(()(

where

)()(

)()(

tFtxtqt

tt

tFtF

iii

i

i


Orientation and rotational movementLinear momentum

Linear momentum p of a particle with mass m and velocity v:

Linear momentum of a rigid body


)(

)(

))()(()()(

))()(()()(

)()(

tvM

tvm

txtrmtwtvm

txtrtwmtvm

trmtP

i

iii

iii

ii

)()( tmvtP

Total linear momentum of a rigid bodyis the same as if the body is simply aparticle with mass m and velocity v(t).

Orientation and rotational movement Linear momentum

Derivative of linear momentum and force

Instead of v(t), we use P(t) as a state variable for the rigid body. Be more consistent with the way we deal with

angular velocity and acceleration.


)()()()(

:constant is

tFtmatvmtP

M


Orientation and rotational movementAngular momentum (book)_

Angular momentum of a particle


)()()(

is

momentumangular its ,)( velocity and , mass

,)( vector local with object, on the particle aFor

trmtrtL

trm

tr

Orientation and rotational movement Angular momentum (book)

Angular momentum of an object


)()(

})])()(())([({

})])(()([))({

}))]()(()([))({

))()(())()((

)()()(

:object theof momentumangular Total

ttL

qtRtwqtRm

qtRtwmqtR

txtqtwmqtR

tvtqmtxtq

trmtrtL

i

i

ii

iii

iii


Orientation and rotational movementAngular momentum

Angular momentum of a rigid body Similar to linear momentum

where I(t) (called inertia tensor) is a 3x3 matrix that describe how the mass in the body is distributed relative to the body’s center of mass.

I(t) depends on orientation of the body; but does not depend on the body’s translation


)( )()( twtItL

Orientation and rotational movementInertia tensor

At a given time t, let be the displacement of particle i from body’s mass center x(t) by

.

I(t) is expressed in terms of r’i as the symmetric matrix


)()()(' txtrtri

)(' tri


How to compute I(t) for a rotated dody? Compute at run-time? Too expensive! Pre-computed I(t) in body-space coordinates and

then compute it at run time after rotation.


Tbody tRtItRtI )()()()(



asrewritten becan )( , Using2,2,2,,, tIrrrrr iziyixi

T

i

:itself with ofproduct outer theTaking ,ir



asrewritten becan )(tI

,)()( and )()( where)()()( Since 0,

0 ItRtRrtRtrtxrtRtr Tiiii



as )( rearrangecan wescalar, a is Since 00 tIrr iTi

Let

We have

Constant over the simulation


Linear momentum and linear velocity

Angular momentum and angular velocity

I(t): inertia tensor, describes how the mass is distributed about the center of mass in space

For an un-transformed object: Iobject

For a transformed object (depends only on orientation)


)( )( tvMtP

)( )()( twtItL

Tobject tRtItRtI )()()()(


Motion Equation


The state of a rigid body Spatial information

Position Orientation

Velocity information Linear momentum Angular momentum

Known information Mass Body-space inertia

Motion Equation


)()()(

)()()(

)()(

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Bodies in collision


Objects moving in a dynamic scene Objects in collision – need to do the detection Objects sliding against on other object Objects resting on each other

Need to calculate forces for computing accurate reaction

Ultimate goal in VR Real and virtual objects need to behave like real

ones. At a minimum

Objects should not pass through each other, and things should move as expected when pushed, pulled, or grasped.

Overall Physical simulation in VR must run reliably,

seamlessly, automatically, and in real time.

Bodies in collision

Animation (U) Chap 7 Physically Based AnimationAnimation (U) Chap 7 Physically Based Animation CS Dept., NCTU, J. H. Chuang

111

Collision detectionComputer Animation


112

Collision detectionHaptic Interaction

From J.C. Latombe’s notes


113

Collision detectionMotion/Path Planning

From James Kuffner

From J.C. Latombe


114

Collision detectionCrowd Simulation

From J.C. Latombe’s notes


To prevent interpenetration Collisions must be detected. Velocities must be adjusted in response to

collisions. Collision response must be computed.

If the collision response does not cause the objects to separate immediately, contact forces must be calculated and applied until separation finally occurs.

Bodies in collision


A naive collision detection has Fixed-timestamp weakness All-pair weakness Pair-processing weakness

Features that current methods have Interactive rate Handle polygon soaps Models can undergo rigid-body motion Provide well-fit bounding volume Collision detection only occurs at discrete times.

Problems of Collision Detection


All-pair weakness If the scenario contains n moving objects and m

static objects, total object test for each frame is

Pair processing weakness Note:

Performance evaluations for collision detection are extremely difficult since algorithms are sensitive to the actual scenarios, and there is no algorithm that performs best in all cases.

Performance of a naive method


2

nnm

Bounding volume for object level and polygon level. Pure bounding volume Hierarchical bounding volumes

Support progressive collision detection.

Spatial coherence Usually large regions of the space are occupied by only one object or

none at all. If some objects share the same region, they are likely intersecting.

Time coherence Moving objects usually move on a continuous path. Results of earlier

collision query can be exploited.

Speed Up Methods


Two-level Collision Detection

Globally searching pairs of objects that have

potential collision Spatial partition

Dimension reduction and sorting (Sweep-and-Prune) Collision detection on a pair of objects

Hierarchical CD based on OBBTree, AABBTree,… Polygon-polygon overlap test

Two-level Collision Detection

Animation (U) Chap 7 Physically Based AnimationAnimation (U) Chap 7 Physically Based Animation CS Dept., NCTU, J. H. Chuang119 119

CS Dept., NCTU, J. H. Chuang

Two-level CD system


Simulation

Collision response

Pruning Multi-Body pairs(Sweep/Prune)

Exact collisionDetection (OBB tree)

Spatial coherence is usually exploited by partitioning the space. All-pair weakness can be resolved by the use of

spatial coherence. Space partition methods differ in fast neighbor-

finding (for static environments) and fast updating for moving objects. Grid (uniform partition) Octree Binary space partition

Space Partition


Sorting objects in 3-D space by dimension reduction If two objects collide in a 3D space, their orthogonal

projections onto the xy, yz, and xz-planes and x, y, and z-axes must overlap.

Project bounding volume of objects. Based on Axis-aligned bounding boxes (AABB).

Fixed-size bounding cube: large enough to contain the object at any orientation.

Dynamically-resized rectangular bounding boxes

Temporal Coherence One or Two-Dimension Sweep and Prune


Project each 3D box onto x, y, and z axes. Construct 3 lists, one for each dimension.

Each list contains the values of the endpoints of the projected interval corresponding to that dimension.

Determine which intervals overlap by sorting these lists. Utilize temporal or frame coherence by changing

only the values of the interval endpoints. Bubble sort or insertion sort work well for

previously sorted lists.

One or Two-Dimension Sweep and Prune


Fixed-sized cubes are preferred over dynamically-resized boxes, except for the class of oblong objects.

Which sorting method is better? Bubble sort works better for environments where

only a few objects move, such as walkthrough Insertion sort works better for environments where

large number of objects move. Overlap status consists of a Boolean flag for

each dimension. Whenever all three of these flags are set, the bb of objects pair overlap.

One or Two-Dimension Sweep and Prune Implementation issues - -1


Bounding volume is used to do a pre-check before doing any further collision detection. Bounding sphere (BS)

Fast overlapping check, not a tight fit. Axis-aligned bounding box (AABB)

Fast overlapping check, a better tight fit. Oriented bounding box (OBB) Tight fit,

but with relatively expensive overlapping test. K-DOPS (direct oriented polytopes)

OBB + more cutting axes

Bounding Volumes



Bounding Volumes


Hierarchical bounding volume is useful For efficiently resolving the pair-processing

weakness, and Providing a basis for the hierarchical or time-

critical scheme for collision detection. Three types:

Hierarchical Axis-aligned bounding box Hierarchical Oriented bounding box Hierarchical Bounding sphere

Hierarchical Bounding Volumes


Common denominators of these schemes A hierarchical bounding volume is built for each

object. High-level code for a collision query is similar,

regardless of the BV type used. BV-BV overlapping tests and primitive-primitive

overlapping tests are different depending on what BVs and primitives are used.

A simple cost function can be used to trim, evaluate, and compare performance.

General Hierarchical Collision Detection Scheme


Common hierarchy used: K-ary tree, where each node may at most have k

children. At each internal node, there is a BV enclosing all its

children. At each leaf, there are one or more primitives.

Three ways of building a hierarchy Bottom-up Incremental tree insertion Top-down

Hierarchical Collision Detection Hierarchy Building


Start Starts by combining a number of primitives and

finding a BV for them. Grouping

This BV is grouped with one or more BVs constructed in a similar way, thus yielding a new, larger parent BV.

Recursive grouping Repeat the grouping until only one BV exists,

which becomes the root of the hierarchy.

Hierarchical Collision Detection Bottom-

up Hierarchy Building


Start with an empty tree All other primitives and their BVs are added one at a time to

the tree. To make an efficient tree, an insertion point should be selected so that

the total tree volume increase is minimized.

Little is known about this scheme in the context of collision detection.

Hierarchical Collision Detection Hierarchy Building by Incremental tree-insertion


Start Find a BV for all primitives of the object, which is then acts as the root

of the hierarchy.

Recursively apply a divide-and-conquer strategy First to split the BV into k or fewer parts and find all included

primitives for each such part. A BV is then created for each part.

Potential advantage A hierarchy can be created on an as-needed basis, i.e., we can construct

the hierarchy for those parts of the object where it is actually needed. Used by majority of hierarchical CD.

Hierarchical Collision Detection Top-Down Hierarchy Building


K-ary tree K =2 minimizes the work to be done when

traversing a path from root to the worst-case leaf, and it is easier to compute the hierarchy than the case of higher k.

Higher k gives a tree with lower height, but it requires more work at each node.

Hierarchical Collision Detection


Challenges Find tight-fitting bounding volume. Find hierarchy construction that creat balanced and

efficient tree. Efficient BV-BV overlapping test. Efficient primitive-primitive overlapping test.

Notes Balanced trees are expected to perform best in all cases,

since the time of CD query will not vary. But, it does not mean that it is best for all inputs, e.g.,

those parts that seldom or never be queried for a collision can be located deep in the hierarchy.

Hierarchical Collision Detection


Collision detection Test whether or not two objects collide, and may

terminate whenever a pair of triangles has been found to be overlapping.

Collision determination In some cases, all pairs of overlapping triangles

might be wanted. Can be solved with small alterations of the code

for collision detection.

Hierarchical Collision DetectionCollision testing between Hierarchies -1


FindFirstHitCD(A, B) return({TRUE, FALSE});

if (not-overlap(ABV, BBV) return FALSE; else if(isLeaf(A)) /* try to reduce tested pairs */ if(isLeaf(B))

for each triangle pair TA in AC and TB in BC

if(overlap(TA, TB)) return TRUE;

else for each child CB in BC

FindFirstHitCD(A, CB)

else for each child CA in AC

FindFirstHitCD(CA, B) return FALSE;

Hierarchical Collision DetectionCollision testing between Hierarchies -2



Hierarchical Collision DetectionCost function -1

Cost function


BV a updatingfor cost :

time)-runat resized is BV the(if

motion smodel' todue updated BVs ofnumber :

testoverlap primitiveprimitive ofcost :

testsoverlap primitiveprimitive ofnumber :

testoverlap BVBV ofcost :

testsoverlap BVBV ofnumber :

where

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Conflicting goals Creating a better hierarchical representation results

in lower values of nv, np, and nu. Creating better methods for BV-BV or primitive-

primitive tests would lower cv, cp. Bounding volume examples

Sphere, AABB, OBB K-DOPS (direct oriented polytopes) Pie slices Spherical shells: good fits for Bezier surfaces

Hierarchical Collision DetectionCost function -2


Why OBBTree? Sphere tree and AABBTree do very well in

performing rejection tests, whenever two objects are far apart; but do poorly for objects in close proximity and having multiple contacts.

OBBTree was designed to perform especially well if parallel close proximity is found during collision detection.

Parallel close proximity: Two surfaces are very close and nearly parallel.

A fast overlapping test for OBB-OBB is provided.

Hierarchical OBBTree


Collision detection based on OBBTree Preprocessing

Building an OBBTree for each object. Top-down Bottom-up

Detection computation Perform recursively the separating-plane test for

each pair of objects.

Hierarchical OBBTree


Split the polygon soap along the longest axis of the OBB.

Pass through the OBB’s center OBBTree may not balanced. If the split fails, the next longest axis can be used.

Pass the point that corresponding to the median of the projection of all triangle centroids onto this axis.

OBBTree can be balanced.

Partition the polygons into two sub-groups based on their centroids.

After split, an OBB is found for each sub-group.

Building an OBBTreeTop-down approach


Building an OBBTreeTop-down approach


For a given triangle soap, compute an orientation of the triangles vertices (via the first and second order statistics): = average of all vertices C = the covariance matrix

Based on C The normalized orthogonal eigenvectors of C are

used as a basis - axes of the OBB. External vertices along each axis yields sizes of the

OBB.

Building an OBBTreeDerive a good-fit OBB -1


OBB for a polygon soap can also be found by computing an orientation of the triangles of the convex hull

The OBB will be less dependent on the distribution of polygons. The convex hull of an object is computed.

Can be done by, e.g., Quickhull algorithm. In return, we have n triangles, and based on which a covariance matrix

is derived. Centroid of the convex hull Covariance matrix

Normalized eigenvectors of C as the OBB’s axes. Min-max extent on each axis.






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In OBBTree, each OBB is stored with a rigid-body transformation (a rotation R and a translation vector t ) matrix MA.

When testing CD of OBBs A and B, The overlap test should be done in the coordinate

system of one of OBBs, say A. So A is now a AABB. Transform B into A’s coordinate system with

Fast Overlap Test for OBBsHandling Rigid-Body Motions


BAAB MMT 1

A naïve algorithm requires 144 edge-face test (12 edges x 6 faces x 2 boxes)

Separating axis theorem For any two arbitrary, convex, disjoint polyhedra, A and B, there exists

a separating axis where the projections of A and B are also disjoint. Checking for disjoint OBBS involves searching for a separating axis. If A and B are disjoint, then they can be separated by a separating axis

that is orthogonal to either A face of A, or A face of B, or An edge from each polyhedron.

Fast Overlap Test for OBBsSeparating Axis Theorem




Axes p and q are orthogonal to faces of A.Axes s and t are orthogonal to faces of B.It is sufficient to find one axis that separates the projection inorder to know that the OBBs do not overlap.Here, q is the only axis that separates the projections.


General separating-plane test: 15 tests Three axes of A, Three axes of B, then Nine axes from A and B (9 axes).

Sequentially test for each axis candidate Do axis projection and form the interval on the axis. Two objects are disjoint if their intervals don't overlap. If the intervals overlap, boxes may or may not be disjoint. Further tests

on other axes may be needed.

More efficient way? What are the axes and in which order?




Fast Overlap Test for OBBs A fast separation test -1

Assume that a potential separating axis is l. Radii of the OBBs A and B on the axis l


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Fast Overlap Test for OBBsA fast separation test -2

Done in the coordinate system of A Formed by A’s center and axes

ac, axes au, av , aw

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It is sufficient to find one axis that separates A and B to be sure that they are disjoint.

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If any of these 15 tests is positive, the OBBs are disjoint.

Number of operations Maximum 180, or 240 if transformation of B into A’s

coordinate system in included. Order of axes

Has an impact on performance. First test three axes of A

They are orthogonal and thus reject the overlap faster, and the simplest tests.

Then three axes of B, followed by the axes formed by axes of A and B.



Fast Overlap Test for OBBs Another fast separation test -7

Simply skip the last 9 axis tests can speed up the overlapping test, Geometrically, this is amount to doing two

AABB-AABB tests. But sometimes report two disjoint OBBs as

overlapping. In these cases, the recursion in the OBBTree

will go deeper than necessary.



Two-level CD system

Globally searching pairs of objects that have potential collision. Spatial subdivision Sweep-and-Prune

Collision detection on a pair of objects Hierarchical CD based on OBBTree,

AABBTree,… Polygon-polygon overlap test



Two-level CD system


Simulation

Collision response

Pruning Multi-Body pairs(Sweep/Prune)

Exact collisionDetection (OBB tree)


Time-critical collision detection

Time critical computing needs to consider all computations involved in each frame update.

Two major computations Rendering and Collision detection

Time-critical collision detection Progressive refinement on the discrete-time collision

detection using hierarchical representations, such as Sphere tree, box tree (axis-aligned or oriented).

Hard to estimate the allocated time.


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