IntroductionToSoftBodyPhysics-Skeel

7/28/2019 IntroductionToSoftBodyPhysics-Skeel

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Introduction toSoft Body Physics

SkeelKeng-SiangLee

[email protected]

http://cg.skeelogy.com
mailto:[email protected]://cg.skeelogy.com/http://cg.skeelogy.com/http://cg.skeelogy.com/mailto:[email protected]


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i

Contents

Introduction........................................................................................................................................

1

BaseCodes.......................................................................................................................................... 4

Chapter1 SpringSimulation.............................................................................................................. 7

1.1PhysicsSimulationAlgorithm............................................................................................................ 8

1.2SimulationObjectClasses................................................................................................................. 9

1.2.1SimObjectAbstractClass............................................................................................................ 9

1.2.2SimModel

..................................................................................................................................

12

1.3ForceGeneratorClasses.................................................................................................................. 13

1.3.1ForceGeneratorInterface......................................................................................................... 13

1.3.2Gravity...................................................................................................................................... 14

1.3.3Medium.................................................................................................................................... 15

1.3.4Spring........................................................................................................................................ 17

1.4IntegratorClasses............................................................................................................................ 21

1.4.1IntegratorAbstractClass.......................................................................................................... 21

1.4.2ForwardEulerIntegrator.......................................................................................................... 23

1.5SimulationClass............................................................................................................................... 24

1.6UsingTheNewClassesToCreateASpringSimulation................................................................... 28

Chapter

2

Cloth

Simulation

.............................................................................................................

34

2.1Constraints...................................................................................................................................... 35

2.1.1ConstraintInterface.................................................................................................................. 36

2.1.2LengthConstraint..................................................................................................................... 37


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ii

2.1.3PointConstraint........................................................................................................................ 39

2.1.4ProcessingTheConstraintsInTheSimulationClass................................................................ 40

2.2Verlet(NoVelocity)Integrator........................................................................................................ 42

2.3SimulationVertex............................................................................................................................ 44

2.4SoftBodyandClothSimulationClasses.......................................................................................... 45

2.5UsingTheNewClassesToCreateAClothSimulation.................................................................... 53

Chapter3 GoalSoftBodies:SlinkySimulation................................................................................. 59

3.1GoalSoftBodySimulationClass...................................................................................................... 60

3.2CreatingAGoalSoftBody............................................................................................................... 63

Chapter4FakingRigidBodies:ChainSimulation............................................................................ 70

4.1ChainSimulationClass.................................................................................................................... 70

4.2CreatingAChainSimulation............................................................................................................ 73

Conclusion........................................................................................................................................ 77

References........................................................................................................................................ 78


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Introduction

Inrecentyears,therehasbeenahighproliferationintheusageofphysicssimulationsingames.This

might

be

due

to

the

increasing

need

for

realistic

movements

of

the

objects

to

match

their

realistic

renderings.Someofthesephysicssimulationsincluderigidbodysimulationswheretheobjectsmove

androtatebutdonotchangetheirshapes,andragdollphysicswherethemotionofacharactercanbe

simulatedunderforceperturbations. Therehashowevernotbeenalotofsoftbodyphysicsingames

yet, thus itmight beworthwhile to start diving into this type of simulation so that you can start

implementingsomesoftbodiesforyourgame.

Asoftbody isbasicallyanobjectwhichchanges itsoverallshapeduetoexternal forcesactingon it.

Someexamplesthatwecommonlyseeincludecloth,balloonsandjelly.Aclothdrapesdownwardsdue

togravitationalforcesandfluttersduetowindforces.Similarly,aballoonenlargeswhentheinternal

pressureforces

increase,

and

produces

indentations

when

achild

exerts

forces

on

it

by

squeezing.

Thesekindofnaturalbehaviorsareinterestingtoobserve,especiallyinagameenvironmentwherethe

entertainmentvalueisofhighimportance.

This tutorial has been written to help you get a quick understanding of some basic theories and

implementationsbehindsimulatingsoftbodies inrealtime. It ismeantforbeginnersaswewillstart

fromtheverybasics. Inanefforttoshowtheusefulnessofsoftbodyphysics,someoftheexamples

giveninthistutorialalsoillustratehowitcanbeusedasaquicksubstitutionforcaseswhereusingrigid

bodysimulationwillprovetobemuchmoreexpensive.

Thereare

four

major

chapters

in

this

tutorial

and

we

will

be

creating

adifferent

soft

body

simulation

foreachofthem.Thistutorialprogressivelybuildsanobjectorientedsoftbodyphysicssystem,thus

wewillbeheavilyreusingandinheritingfromthecodeswritteninearlierchapters.Itassumesthatyou

haveabasicunderstandingofobjectorientedprogramming.

This tutorialprimarily targetsusersofMicrosoftXNA3.0but the ideasandalgorithmscaneasilybe

implementedusingotherlanguagesandAPIsonceyouunderstandthem.

Bytheendofthiswholetutorial,youwillbeabletocreatethesesimulations:

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Video1:Aninteractiveclothsimulation[Clickonimageabovetowatchthevideo]

Video2:

An

interactive

slinky

simulation

(created

using

a

goal

based

soft

body)

[Clickonimageabovetowatchthevideo]

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Video3:Aninteractivechainsimulation(createdusingspringsandlengthconstraints)[Clickonimageabovetowatchthevideo]

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BaseCodes

Ihave includedsomebasicgamecomponentsandclasses intheprojectfilesofeachchaptersothat

certainbasictaskscanbeperformedeasily.Someofthethingsthatthesegamecomponentscando

include:

i) LoadinganddrawingamodelfromfilebycallingLoadModel ( t heFi l eName) ii) TransformingofamodelinthescenebysettingtheTr ansl at e,RotateandScal epropertiesiii) CreatingofacamerabycreatingaCamer aComponent andaddingittogame. Component siv) Detecting if a key is held down, just pressed or just released by calling the methods

I sKeyHel dDown( ) ,I sKeyJ ust Down() ,I sKeyJ ust Up( ) respectively

v) Creatinga3DlinebyjustcreatingaLi ne3DComponent ,addingittogame. Component s andthensettingtheSt art Pos i t i on,EndPosi t i onandCol or properties

These base codes basically encapsulates some of the basic operations in XNA. They aremeant to

improvetheclarityofthesoftbodysimulation implementations,whichwillotherwisebeaffectedby

unnecessarycodesthatarethereto loadmodels,drawprimitives,detect ifakey isjustpressedetc.

TheseclassesareprettymuchthebasicsofXNA,soifyouknowyourXNAwellenough,itshouldnotbe

toohardtounderstandwhatiswrittenintherewithoutmeexplainingthem.

Touse thebasecodes,youjustneed tocreateanew instanceofeachof thecomponentsandadd

themtothet hi s. Componentslistinyourgame.ThishasbeendoneforyouinGame1.csineachofthe

chapterfiles.

( Game1. cs, addi t i on of codes i n bol d)

usi ng Mi cr osof t . Xna. Framework;usi ng Mi cr osof t . Xna. Framework. Gr aphi cs;usi ng Mi cr osof t . Xna. Framework. I nput ;usi ng Skeel Sof t BodyPhysi csTut ori al . Mai n;

namespace Skeel Sof t BodyPhysi csTut or i al{

publ i c c l ass Game1 : Mi cr osof t . Xna. Framework. Game{

Gr aphi csDevi ceManager gr aphi cs;Spr i t eBat ch spr i t eBat ch;

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/ / component member sCamer aComponent camer aComponent ;I nputComponent i nput Component ;Model Component model Component ;Li ne3DComponent l i ne3DComponent ;

publ i c Game1( ){gr aphi cs = new Gr aphi csDevi ceManager ( t hi s) ;Cont ent . Root Di r ect or y = "Cont ent " ;

/ / i nst ant i ate t he component s and add t hemt o t hi s. Component sVect or3 cameraPosi t i on = new Vect or3( 0, 10, 10) ;Vect or3 t ar get Posi t i on = Vect or3. Zer o;Vect or3 upVect or = Vect or3. Uni t Y;st r i ng backgr oundI mage = "" ;camer aComponent = new Camer aComponent ( t hi s, camer aPosi t i on,

t arget Posi t i on, upVect or, backgr oundI mage) ;t hi s. Component s. Add(camer aComponent ) ;

bool showMouse = f al se;i nput Component = new I nputComponent ( t hi s, showMouse) ;t hi s. Components. Add( i nputComponent ) ;

model Component = new Model Component ( t hi s) ;t hi s. Components. Add( model Component) ;

l i ne3DComponent = new Li ne3DComponent ( t hi s) ;t hi s. Components. Add( l i ne3DComponent ) ;

}

pr ot ect ed over r i de voi d I ni t i al i ze( ){

base. I ni t i al i ze( ) ;}

pr ot ect ed over r i de voi d LoadCont ent ( ){

spr i t eBat ch = new Spr i t eBat ch( Gr aphi csDevi ce) ;}

pr ot ect ed over r i de voi d Unl oadCont ent ( ){

}

pr ot ect ed over r i de voi d Updat e( GameTi me gameTi me){

base. Updat e( gameTi me) ;}

pr ot ect ed over r i de voi d Dr aw( GameTi me gameTi me){

Gr aphi csDevi ce. Cl ear ( Col or. Cor nf l ower Bl ue) ;base. Dr aw( gameTi me) ;

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}}

}

Therestofthecodesabovearejustthestandardtemplatecodesthatareprovidedwhenyoucreatea

newprojectinXNAGameStudio,withthecommentsremoved.

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Chapter1 - SpringSimulation

GetthesourcefilesforthestartofthischapterfromtheSkeelSoftBodyPhysicsTutorialChapter1BEGINfolder

Wewill

start

off

the

tutorial

by

implementing

asimple

spring

simulation

where

two

objects

are

attachedviaaspring.By learninghowtocreateaspringsimulation,youwillgainenoughknowledge

onhow toperformsoftbodysimulations ingames.Thissimplesimulationwillthusprepareyou for

morecomplicatedsoftbodysimulationsinlaterchapters.

Bytheendofthischapter,youshouldhaveasimplespringsimulationlikethis:

Video4: Asimplespringsimulationbetweentwoobjects[Clickonimageabovetowatchthevideo]

Itdoesnotlookparticularlyimpressivebecauseitisjustabasictestofhowtocreateaspringandrun

the simulation. However, do not get too worried about it as we will create much betterlooking

simulationsin

later

chapters.

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1.1PhysicsSimulationAlgorithmToimplementanyphysicssimulationingames,weneedtodothesesteps(inorder):

i)Sum

up

all

forces

acting

on

a

body/vertex

to

get

the

resultant

force

ii) Findaccelerationbydividingtheresultantforcebytheobjectmass:a=f/m(NewtonsSecondLawforaconstantmass)

iii) Performnumericalintegrationontheaccelerationtogetvelocity/positioniv) Updatethebody/vertexwiththenewvelocity/position,andrepeatfrom(i)

Ateach time step,wegetanewposition for the simulatedobject,basedonall the forces thatare

actingon it.Asweupdate theobjectwith thisnewposition,wewillobserve itsmotionunder the

simulation.

Wewillcovermorealgorithmsandformulasaswemovealongwiththeimplementation,particularly

onhowthedifferenttypesofforcesarecalculatedandhowtoperformthenumericalintegration.

Inorder to implement thissimulationsystem, thereare fourmaingroupsofclasses thatwewillbe

creating:

Simulationobjects:Thesepackanobject(canbeamodel,vertex,triangleetc)togetherwithitssimulation

attributes

such

as

position,

velocity

and

mass

into

one.

Forcegenerators:Theserepresentthingsthatcangenerateforces,suchasaspringorgravity. Integrators: These perform numerical integration using the acceleration to find the new

position/velocity,andupdatesthecorrespondingattributesinthegivensimulationobject.

Simulations:Thesearethemainclassesthatwillmanagethewholesimulationalgorithm.Thetasksthattheywilldoincludeinitializingallthevariables,connectingthespringsandupdating

thesimulations.

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1.2SimulationObjectClassesRecallthatweneedtokeeptrackofthepositionandpossiblythevelocityof theobjectundergoing

simulation. It will be convenient if we pack the simulated object together with these simulation

attributes.Thus,

let

us

first

start

with

the

simulation

object

classes.

1.2.1SimObjectAbstractClass

WefirstcreateanabstractclasscalledSi mObj ect .Thiswillcontainnecessarymembersandfunctions

thatwillbeinheritedbyothersimulationobjectclasses.

UsingtheSolutionExplorer,addanewfoldercalledSimObjects intheexistingSoftBodyfolder.Then

createanewclassfileandnameitSimObject.cs.

( Si mObj ect . cs)

usi ng Mi cr osof t . Xna. Framework;usi ng Skeel Sof t BodyPhysi csTut ori al . Mai n;

namespace Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Si mObj ect s{

publ i c enum Si mObj ect Type { PASSI VE, ACTI VE }

publ i c abst r act c l ass Si mObj ect

{ pr i vat e f l oat mass;pr i vat e Si mObj ect Type si mObj ect Type;pr ot ect ed Vect or3 cur r Posi t i on;pr ot ect ed Vect or3 pr evPosi t i on;pr ot ect ed Vect or3 cur r Vel oci t y;

publ i c f l oat Mass{

get { return mass ; }set { mass = val ue; }

}

publ i c Si mObj ect Type Si mObj ect Type{

get { return si mObj ectType; }set { si mObj ect Type = val ue; }

}

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publ i c Vect or3 Cur r Posi t i on{

get { return cur r Posi t i on; }set { cur r Posi t i on = val ue; }

}

publ i c f l oat Cur r Posi t i onX

{ get { return cur r Posi t i on. X; }set { cur r Posi t i on. X = val ue; }

}

publ i c f l oat Cur r Posi t i onY{

get { return curr Pos i t i on. Y; }set { cur r Posi t i on. Y = val ue; }

}

publ i c f l oat Cur r Posi t i onZ{

get { return cur r Posi t i on. Z; }

set { cur r Posi t i on. Z = val ue; }}

publ i c Vect or3 Pr evPosi t i on{

get { return pr evPosi t i on; }set { pr evPosi t i on = val ue; }

}

publ i c Vect or3 Cur r Vel oci t y{

get { return cur r Vel oci t y; }set { cur r Vel oci t y = val ue; }

}

/ / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

publ i c Si mObj ect ( f l oat mass, Si mObj ect Type si mObj ect Type){

t hi s. mass = mass;t hi s. curr Pos i t i on = Vect or3. Zer o;t hi s. pr evPosi t i on = cur r Posi t i on;t hi s. cur r Vel oci t y = Vect or3. Zer o;t hi s. si mObj ect Type = si mObj ect Type;

}}

}

ASi mObj ect hasamassandisofaSi mObj ect Typewhichiseitheractiveorpassive. Activesimulation

objectshavetheirpositionsandvelocitiesdeterminedbythe forces inthesystem,whilethepassive

onesdonot. Passive simulationobjects are still consideredpartof the simulation since they allow

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activeobjectstointeractwiththem.Settinganobjecttobepassiveisusefulincaseswhereweneedto

allowtheusertocontrolitsposition,possiblyusingthemouseorkeyboard.

In the constructor above, we also initialize the curr Pos i t i on, pr evPosi t i on and cur r Vel oci t y

variablestothezerovector.

Recallthat

we

have

to

sum

up

the

forces

acting

on

this

simulation

object.

It

will

be

convenient

for

the

object itselftostoretheresultantforcesothattheobjectcanbepassedaroundtodifferentclasses

and the resultant force is still being tracked and stored. Thus we will create a r esul t ant For ce

variableasaclassmemberintheSi mObj ect .

( cl ass member i n Si mObj ect . cs)

pr ot ect ed Vect or3 r esul t ant For ce;

publ i c Vect or3 Resul t ant For ce{

get { return r esul t ant For ce; }set { r esul t ant For ce = val ue; }

}

Wewillalsocreateamethodtoresettheresultantforce.Thiswilltypicallybecalledeitherattheend

oratthebeginningofeachtimestep.

( cl ass met hod i n Si mObj ect . cs)

publ i c voi d ResetFor ces( ){

t hi s. r esul t ant For ce = Vect or3. Zer o;}

Finally,anyclassthatinheritstheSi mObj ect classshouldhaveitsownUpdat emethod,sowedefine

anabstractmethodcalledUpdat etomakeitcompulsoryforderivedclassestoimplementit.

( cl ass abst r act met hod i n Si mObj ect . cs)

publ i c abst r act voi d Updat e( GameTi me gameTi me) ;

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1.2.2SimModel

Based on the Si mObj ect abstract class,we shall implement our first simulation object class called

Si mModel .Itismeanttowrapsimulationattributestoamodelthatwewillloadfromfileandupdate

thepositionofthisloadedmodelateverytimestep.IntheSimObjectsfolder,createanewclassfile

namedSimModel.cs.

( Si mModel . cs)

usi ng Mi cr osof t . Xna. Framework;usi ng Skeel Sof t BodyPhysi csTut ori al . Mai n;


publ i c seal ed cl ass Si mModel : Si mObj ect{

pr i vat e GameModel model ;

publ i c GameModel Model{

get { return model ; }set { model = val ue; }

}

/ / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

publ i c Si mModel ( GameModel model , f l oat mass, Si mObj ect Type si mObj ect Type): base( mass, si mObj ect Type)

{t hi s. model = model ;

t hi s. cur r Posi t i on = model . Transl at e;t hi s. pr evPosi t i on = cur r Posi t i on;}

}}

Basicallywe pass in a GameModel (oneof the classesprovided in thebase codes), its mass and its

Si mObj ect TypewhenwecreateaSi mModel instance.Thecur r Posi t i onandpr evPosi t i onvariables

areupdatedtothecurrentpositionoftheGameModel .

Sincethe

Updat emethod

is

defined

as

abstract

in

the

base

class,

we

will

have

to

implement

it

using

an

override method. All we have to do for this method is to update the position of the GameModel

according to the cur r Posi t i on, which will be recalculated at every time step. This basically

synchronizesthepositionoftheGameModel withthestoredposition.

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( cl ass method i n Si mModel . cs)

publ i c over r i de voi d Updat e( GameTi me gameTi me){

t hi s. model . Transl at e = t hi s. cur rPos i t i on;}

Thatisallwehavetodoforthisclass,sincetheothernecessaryimplementationsareallinheritedfrom

thebaseclass.

1.3ForceGeneratorClassesNow thatwehave the simulationobjectscreated,we can start to implement the forcegenerators.

Recallfrom

the

simulation

algorithm

in

Section

1.1

that

we

need

to

sum

up

the

forces

that

are

acting

onabodyateverytimestep.Theseforceswillcomefromtheforcegeneratorclassesthatwewillbe

implementingnow.

1.3.1ForceGeneratorInterface

WeshallstartoffbycreatingaForceGenerat or interfaceandmakeitcompulsoryforderivedclasses

toimplementamethodAppl yFor ce.

Createanew

folder

called

ForceGenerators

in

the

SoftBody

folder,

and

create

anew

class

file

named

ForceGenerator.csinthenewfolder.

( For ceGenerat or. cs)

usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Si mObj ect s;

namespace Skeel Sof t BodyPhysi csTut ori al . Sof t Body. For ceGener at or s{

publ i c i nt erf ace ForceGenerat or{

voi d Appl yFor ce(Si mObj ect si mObj ect ) ;}

}

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WhatweareexpectingthederivedclassestodowhenimplementingtheAppl yFor cemethodistouse

thepropertiesoftheSi mObj ect (suchasmass,position,velocityetc)tocalculateanappropriateforce

andaddthattoitsResul t ant For ceproperty.

1.3.2Gravity

Letusproceedon tocreateour firstandsimplest forcegenerator:gravity. Toget thegravitational

forceactingonanobject,weneedtwopiecesof information,namelythemassofthebodyandthe

gravitationalaccelerationthatitisexperiencing.Thegravitationalforcecanthensimplybefoundas

gravitationalforce=mass*gravitationalacceleration [1]

Now,addanewclassfilenamedGravity.csintheForceGeneratorsfolderandcreateaGr avi t yclass

whichinheritsfromtheForceGenerat or interface:

( Gr avi t y. cs)

usi ng Mi cr osof t . Xna. Framework;usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Si mObj ect s;

namespace Skeel Sof t BodyPhysi csTut ori al . Sof t Body. For ceGener at or s

{publ i c seal ed cl ass Gr avi t y : ForceGenerat or{

pr i vat e Vect or3 accel er at i on;

publ i c Vect or3 Accel er at i on{

get { return accel er at i on; }set { accel er at i on = val ue; }

}

publ i c f l oat Accel er at i onX{

get { return accel er at i on. X; }set { accel er at i on. X = val ue; }

}

publ i c f l oat Accel er at i onY{

get { return accel er at i on. Y; }set { accel er at i on. Y = val ue; }

}

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publ i c f l oat Accel er at i onZ{

get { return accel er at i on. Z; }set { accel er at i on. Z = val ue; }

}

/ / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

publ i c Gr avi t y( ): t hi s( new Vect or3( 0, - 9. 81f , 0) ) { }

publ i c Gr avi t y( Vect or3 accel er at i on): base( )

{t hi s. accel er at i on = accel er at i on;

}}

}

The first constructor takes in no acceleration but creates an acceleration of 9.81 downwards by

default.ThisisthegravitationalaccelerationthatweexperienceonEarth.

Wenowhave to implement the Appl yFor cemethod.Asdescribed inEqn.1,weneed to find the

product of the object mass and gravitational acceleration, and add that to its Resul t ant For ce

property.

( cl ass met hod i n Gr avi t y. cs)

publ i c voi d Appl yFor ce(Si mObj ect si mObj ect ){

si mObj ect . Resul t ant For ce += si mObj ect . Mass * accel erat i on;}

1.3.3Medium

Next,weimplementaforcegeneratorcalledMedi umwhichrepresentsthemediumthatthesimulation

objectisin.Theviscosityofthemediumthatthebodyisinwilldeterminehowmuchdragforceitwill

experience.Anobjectmovinginhoney,forinstance,willexperiencemoredragforcesthananobjectin

air.To

simplify

the

implementation,

we

shall

only

represent

the

viscosity

by

adrag

coefficient

that

can

bearbitrarilyadjustedbytheusertogetthedesireddrageffect.

AddaMedium.csfileintheForceGeneratorsfolder.

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( Medi um. cs)



publ i c seal ed cl ass Medi um : ForceGenerat or

{ pr i vat e f l oat dr agCoef f i ci ent ;

publ i c f l oat Dr agCoef f i ci ent{

get { return dr agCoef f i ci ent ; }set { dr agCoef f i ci ent = val ue; }

}

/ / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

publ i c Medi um( f l oat dr agCoef f i ci ent ): base( )

{

t hi s. dr agCoef f i ci ent = dr agCoef f i ci ent ;}

}}

Asusual,wehavetoimplementtheAppl yFor cemethod.Thedragforceisusuallyproportionaltothe

currentvelocityofthebody.Thefaster it ismoving,themoredragforce itexperiences.Thus,allwe

havetodoismultiplyourdragcoefficientwiththecurrentvelocityofthesimulationobject:

dragforce= dragcoefficient*objectvelocity [2]

Sincethisisanopposingforcethatslowsdownthemotionoftheobject,wehavetonegatetheresult.

Theimplementationforthisisasfollows:

( cl ass met hod i n Medi um. cs)

publ i c voi d Appl yFor ce(Si mObj ect si mObj ect ){

si mObj ect . Resul t ant For ce += - dr agCoef f i ci ent * si mObj ect . Cur r Vel oci t y;}

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1.3.4Spring

Wenowimplementthelastforcegenerator,whichisaspring.Aspringhastheseattributes:

Stiffness:Thisdetermines thespringinessof the spring.Thehigher the stiffnessvalue, themoreforcethatwillbegeneratedwhenthespringiscompressedorstretched.Thiscausesthe

springto

return

to

its

original

shape

faster,

thus

creating

astiffer

behavior.

This

is

also

known

as

thespringconstant.

Damping:Thisdetermines theamountof internaldrag force that the springwillexperience.Thedampingforceisparalleltothedirectionofthespring.

Restlength:Thisisthelengththatthespringwilltrytomaintainwhenundergoingcompressionorstretching.

Intermsofimplementation,aspringalsoneedstostorethetwoobjectsarethatattachedtoitsothat

its

current

length

can

be

calculated

at

any

time

step.

Figure1:Springforcesactingonthetwoendsofthespringwhenundergoingcompression

ReferringtoFig.1,thespringforceexperiencedbyobjectAiscalculatedas:

springforceonA= stiffness*(currentlengthrestlength)*unitvectorBA [3]

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Aspringhasitsowndampingforce,inadditiontothedampingforcecausedbythemediumthatitisin.

Thedifferencebetweenthesetwotypesofdampingforcesisthatthespringdampingforceonlyacts

on the spring itselfwhile themediumdamping forcedamps themotionofall theobjects in it.The

springdampingforceisproportionaltotherelativevelocitiesofthetwoobjectsandisusuallywritten

as:

springdampingonA= damping*((velocityofAvelocityofB).unitvectorBA)*unitvectorBA [4]

Note that this damping force should be in the direction from B to A. However, the difference in

velocitiesisavectoranditmaynotbeinthatdesireddirection.Thus,wehavetoprojectthevelocity

differencevectortothedirectionvectorfromBtoA.Thisprojectionisdonebytakingthedotproduct

ofthevelocitydifferencevectorwiththeunitvectorfromBtoA(whichwillgivethemagnitude),and

thenmultiplyingthatwiththeunitvectorfromBtoA(whichwillthengivethevectorthatweneed).

Thereisanegationinthetermbecausethisisanopposingforce.

NewtonsThirdLawofMotionstatesthat forevery forcegenerated,there isanequalandopposite

force.Thus,thespringforceexperiencedbyobjectBisjustthesameforceasexperiencedbyobjectA,

butintheoppositedirection.Thisappliestothespringdampingforceaswell.

Nowthatwehaveenoughknowledgeaboutthespringforces, letusstartto implementtheSpr i ng

class.CreateanewclassfilecalledSpring.csfileintheForceGeneratorsfolder.

( Spr i ng. cs)



publ i c seal ed cl ass Spr i ng : ForceGenerat or{

pr i vat e f l oat s t i f f ness;pr i vat e f l oat dampi ng;

pr i vat e f l oat r estLengt h;pr i vat e Si mObj ect si mObj ect A;pr i vat e Si mObj ect si mObj ect B;

publ i c f l oat St i f f ness{

get { return st i f f ness; }set { st i f f ness = val ue; }

}

18


22/81

publ i c f l oat Dampi ng{

get { return dampi ng; }set { dampi ng = val ue; }

}

publ i c Si mObj ect Si mObj ect A{get { return si mObj ectA; }set { si mObj ect A = val ue; }

}

publ i c Si mObj ect Si mObj ect B{

get { return si mObj ectB; }set { si mObj ect B = val ue; }

}

/ / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

publ i c Spr i ng( f l oat s t i f f ness, f l oat dampi ng, Si mObj ect si mObj ect A,Si mObj ect si mObj ect B)

: t hi s( st i f f ness, dampi ng, si mObj ect A, si mObj ect B,( si mObj ect A. Cur r Posi t i on - si mObj ect B. Cur r Posi t i on) . Lengt h( ) ) { }

publ i c Spr i ng( f l oat s t i f f ness, f l oat dampi ng, Si mObj ect si mObj ect A,Si mObj ect si mObj ect B, f l oat r est Lengt h)

: base( ){

t hi s. s t i f f ness = st i f f ness;t hi s. dampi ng = dampi ng;t hi s. si mObj ect A = si mObj ect A;t hi s. si mObj ect B = si mObj ect B;

t hi s. r est Lengt h = r est Lengt h;}}

}

TheSpr i ngclassbasicallytakesinthenecessaryattributesofaspringandstoresthem.Italsotakesin

thetwoSi mObj ect sthatareattachedtoeachendofthespring. Ifnorest length issupplied tothe

constructor,thenitisjusttakenasthestartingdistancebetweenthetwoSi mObj ect sgiven.

Next,weimplementtheAppl yFor cemethod:

19


23/81

( cl ass met hod i n Spr i ng. cs)

pr i vat e Vect or3 di r ecti on;pr i vat e f l oat cur r Lengt h;pr i vat e Vect or3 f or ce;publ i c voi d Appl yFor ce(Si mObj ect si mObj ect ){

/ / get t he di r ecti on vectordi r ect i on = si mObj ect A. Cur r Posi t i on - si mObj ect B. Cur r Posi t i on;

/ / check f or zer o vect ori f ( di r ect i on != Vect or3. Zer o){

/ / get l engt hcur r Lengt h = di r ect i on. Lengt h( ) ;

/ / nor mal i zedi r ecti on. Nor mal i ze( ) ;

/ / add spr i ng f or cef or ce = - st i f f ness * ( ( cur r Lengt h - r est Lengt h) * di r ecti on) ;

/ / add spri ng dampi ng f orcef or ce += - dampi ng * Vect or3. Dot( si mObj ect A. Cur r Vel oci t y -

si mObj ectB. Cur r Vel oci t y, di r ecti on) * di r ecti on;

/ / appl y t he equal and opposi t e f or ces t o the obj ect ssi mObj ectA. Resul t ant Force += f orce;si mObj ectB. Resul t ant Force += - f orce;

}}

Implementing theAppl yFor cemethod isabitmore involved.We firstget thedirectionvector that

pointsfromobjectBtoA,thenwefinditslengthandnormalizeit.Rememberthatwehavetocheck

whetheravectoristhezerovectorbeforewenormalizeitorelsewewillhaveadivisionbyzero.With

theseinformation,wecancalculatethespringforceandspringdampingforcesaccordingtoEqn.3and

Eqn.4respectively.Finally,weaddtheseforcestoobjectAandaddtheequalbutoppositeforceto

objectB.

NoticethatwearenotusingtheSi mObj ect thathasbeenpassed inasaparameter,sincewehave

alreadystoredboththenecessarySi mObj ect sintheclassitself.Thisisabitofaquirkinesscausedby

implementingthemethodspecifiedintheinterfaceclass.However,itdoesnotcausemuchharmand

wecan

just

pass

in

nul lwhen

we

are

calling

this

method

for

a

Spr i nginstance.

20


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1.4IntegratorClassesRecallthatoncewehavetheaccelerationoftheobject,weneedtoperformintegrationwithrespect

totimetoobtainthevelocityandpositionoftheobject.Inarealtimeenvironment,wearenotableto

performintegration

in

an

analytical

manner,

meaning

that

you

cannot

just

integrate

an

equation

like

what you did in your basic calculus classes (e.g. add one to the exponent and divide thewhole

equationby it).Whatwehavetorelyonnow isanumericalmethodcallednumerical integration.

Suchanumericalsolutionallowsustosampleatdiscretepointsintheintegraltogetanapproximate

solution.Inourcase,wesampletheintegralateachtimestep(forsinglestepintegrators).Numerical

integratorsallowustousesomeequationsinvolvingacceleration,velocityandpositiontoapproximate

thenewpositionofanobject.

One major concept that is associated with numerical integrators is that there is always an error

associatedwith thenumerical integrator thatweuse. This is roughly indicatedby theorderof the

integrator.Thehigher theorder, the lesserror there is.However,higherorder integrationmethods

usuallymeansmorecalculationsandthusslowerinexecution.

There is a variety of integrators available for us to choose from, ranging from a Forward Euler

integrationmethodwhich is the fastestbutnotvery stable (firstorder), tosomething likeaRunge

KuttaFourthOrderwhichisverystablebutveryexpensiveincalculationsaswell.Forgames,thereisa

popularmiddlegroundtechniquecalledVerletintegrationwhichisfastandstable(secondorder).For

thistutorial,weshallstartoffwithaForwardEulerintegratorsinceitisthesimplesttounderstandand

implement,andthenmoveontoaVerletintegratorinlaterchapters.

Theseintegratorsrequiredifferentattributesoftheobject.Somerequireyoutostorethecurrentand

previousposition,whilesomerequireyoutostorethecurrentpositionandcurrentvelocity.Whatever

thecaseis,thenewpositionforthenexttimestepisalwaysgenerated,andthatwillbeusedtoupdate

theCur r Posi t i onpropertyoftheSi mObj ect .

Now thatwe have a brief overview of numerical integrators,we can start creating the Integrator

classes. These classes basically take in the acceleration of a Si mObj ect and performs numerical

integrationtofindthenewpositionandpossiblyvelocity.

1.4.1IntegratorAbstractClass

Asusual,wewillfirstcreateanabstractclassforotherintegratorclassestoderivefrom. First,createa

newfoldercalledIntegratorsundertheSoftBodyfolder,andcreateanewclassfilenamedIntegator.cs

init.

21


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( I nt egrator. cs)


namespace Skeel Sof t BodyPhysi csTut or i al . Sof t Body. I nt egr at or s{

publ i c abst r act c l ass I nt egr at or{pr i vat e Game game;

/ / use a f i xed t i me st ep t o get pr edi ct abl e si m pr ot ect ed f l oat f i xedTi meSt ep;

publ i c f l oat Fi xedTi meSt ep{

get { return f i xedTi meSt ep; }set { f i xedTi meSt ep = val ue; }

}

/ / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

publ i c I nt egr at or ( Game game){

t hi s. game = game;

/ / set t he f i xed t i me st ep t o target el apsed t i me ( def aul t at 1/ 60)f i xedTi meSt ep = ( f l oat ) game. Tar getEl apsedTi me. Tot al Seconds;

}}

}

The

I nt egr at orabstract

class

defines

aclass

variable

called

f i xedTi meSt epwhich

stores

afixed

time

stepvalueforthesimulation. Simulationbehaviorchangesasthetimestepchanges,meaningthata

spring, for instance, with the same stiffness and damping constants will behave differently when

undergoing a simulation at 30 frames per second and at 60 frames per second. In order to get

consistent and predictable simulations, it is better to keep it at a constant value. There are other

solutionsforthis,suchasadaptivetimestepmethods,butwewillkeeptothissimplesolutioninour

tutorial.Forthe f i xedTi meSt epvariable,wewillbeusingthevaluefromgame. Target El apsedTi me

whichspecifiesthedesiredtimestepforthegame.Thedefaultvalueforthisis1/60seconds.

DerivedclasseswillneedtoimplementtheirownI nt egr at emethod,soitisdefinedasabstractinthe

class.

( cl ass abst r act met hod i n I nt egr at or . cs)

publ i c abst r act voi d I nt egr at e( Vect or3 accel er at i on, Si mObj ect si mObj ect ) ;

22


26/81

1.4.2ForwardEulerIntegrator

Wecannowstarttodiscussaboutourfirstnumericalintegrator,calledtheForwardEulerintegrator.

Thisisoneofthemostbasicintegratorthatcanbeimplemented.

Tofindthevalueofanattributeyinthenexttimestep,theForwardEulerintegratorbasicallyfindsthe

productof

the

time

step

tand

the

derivative

of

y,

and

adds

the

result

to

yat

the

current

time

step:

y(t+ t)=y(t)+ y(t)t [5]

Relatingthattooursimulation,wehave:

v(t+ t)=v(t)+a(t) t [6]

Thismeansthatthevelocityatthenexttimestepisequalstothevelocityatthecurrenttimeaddedto

theproductofthetimestepandaccelerationatthecurrenttime.

Asimilarequationexistsforthepositionoftheobject:

x(t+ t)=x(t)+v(t) t [7]

Now,createanewfilecalledForwardEulerIntegrator.cs.

( For war dEul er I nt egr at or . cs)



publ i c seal ed cl ass For war dEul er I nt egr ator : I nt egr at or{

publ i c For war dEul er I nt egr ator ( Game game): base( game) { }

}}

23


27/81

ThisclassbasicallyinheritsfromtheI nt egr at or abstractclass.

We then createanoverridemethodnamed I nt egr at e, anduseEqn.6andEqn.7 given above to

obtain thenew velocityandposition respectively.Thesenewattributes areupdated into the given

Si mObj ect .

( cl ass met hod i n For war dEul er I nt egr at or . cs)

publ i c over r i de voi d I nt egr at e( Vect or3 accel er at i on, Si mObj ect si mObj ect ){

/ / cal cul at e new posi t i on usi ng t he vel oci t y at cur r ent t i mesi mObj ect . Cur r Posi t i on += si mObj ect . Cur r Vel oci t y * f i xedTi meSt ep;

/ / cal cul at e new vel oci t y usi ng t he accel er at i on at cur r ent t i mesi mObj ect . Cur r Vel oci t y += accel erat i on * f i xedTi meSt ep;

}

1.5SimulationClassWenowhaveallthenecessaryclassesthatweneedtoimplementthesimulationalgorithmdiscussed

inSection1.1.Weshallpackthesimulationalgorithm intoaSi mul at i onclassso that it iseasierto

createotherkindsofsimulationsinlaterchapters.

Createanew

folder

called

Simulations

in

the

SoftBody

folder,

and

then

add

anew

class

file

named

Simulation.cs.

( Si mul at i on. cs)

usi ng System. Col l ect i ons. Gener i c;usi ng Mi cr osof t . Xna. Framework;usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. For ceGener at or s;usi ng Skeel Sof t BodyPhysi csTut or i al . Sof t Body. I nt egr at or s;usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Si mObj ect s;

namespace Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Si mul at i ons{

publ i c c l ass Si mul at i on{

pr ot ect ed Game game;pr ot ect ed Li st si mObj ect s = new Li st ( ) ;pr otect ed Li st gl obal For ceGenerators = new

Li st ( ) ;

24


28/81

pr ot ect ed Li st spr i ngs = new Li st ( ) ;pr ot ect ed I nt egr at or i nt egr at or ;

publ i c Li s t Si mObj ect s{

get { return si mObj ect s; }set { si mObj ect s = val ue; }

}

publ i c I nt egr at or I nt egr at or{

get { return i nt egr at or ; }set { i nt egr at or = val ue; }

}

/ / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

publ i c Si mul at i on( Game game){

t hi s. game = game;

/ / creat e a def aul t i nt egr at ort hi s. i nt egrator = new For war dEul er I nt egr at or ( game) ;

}}

}

FromthealgorithmstatedinSection1.1,weknowthatwehavetoiteratethroughlistsofsimulation

objects,forcesandalsouseanintegrator.ThustheSi mul at i onclasskeepstrackof:

A listofSi mObj ect s:Thesearetheobjectsthatareinthesimulationsystem.Theactiveoneshave

their

motion

determined

by

the

simulation

system,

while

the

passive

ones

do

not.

AlistofForceGenerators:Thesearetheglobalforcegenerators,suchasGr avi t yandMedi um.TheywillbeappliedtoallSi mObj ect sinthesystem.

A listofSpr i ngs:TheseareconsideredlocalforcegeneratorsthatwillapplyforceswhichactonlyontheattachedSi mObj ect s.

An integrator: This is used to integrate the acceleration of the Si mObj ect s to obtainpositions/velocitiesforthenexttimestep.

WeshallalsocreatesomemethodstoaddtheSpr i ngs,ForceGenerat or sandSi mObj ect s intothe

simulationsystem:

25


29/81

( cl ass met hods i n Si mul at i on. cs)

publ i c voi d AddSpr i ng( f l oat s t i f f ness, f l oat dampi ng, Si mObj ect si mObj A,Si mObj ect si mObj B)

{Spr i ng spr i ng = new Spr i ng( st i f f ness, dampi ng, si mObj A, si mObj B) ;spr i ngs. Add( spr i ng) ;

}

publ i c voi d AddSi mObj ect ( Si mObj ect si mObj ect ){

si mObj ect s. Add( si mObj ect ) ;}

publ i c voi d AddGl obal ForceGenerat or ( ForceGenerat or f orceGener ator){

gl obal For ceGener ators . Add( f orceGener ator) ;}

Nowwe

create

an

Updat emethod

that

implements

the

physics

simulation

algorithm

in

astep

by

step

fashion.

( cl ass met hod i n Si mul at i on. cs)

publ i c vi r t ual voi d Updat e( GameTi me gameTi me){

}

First,we

sum

up

all

the

forces.

This

is

done

by

iterating

through

each

Spr i ngto

sum

up

the

local

spring

forces,andthenapplyingtheglobalforcestoeachSi mObj ect .

26


30/81

( i nsi de Updat e( ) i n Si mul at i on. cs)

/ / sum al l l ocal f orcesf or each ( Spr i ng spr i ng i n spr i ngs){

spr i ng. Appl yForce(nul l ) ; / / no need t o speci f y any si mObj}

/ / sum al l gl obal f or ces acti ng on t he obj ectsf or each ( Si mObj ect si mObj ect i n si mObj ect s){

i f ( si mObj ect . Si mObj ect Type == Si mObj ect Type. ACTI VE){

f or each ( ForceGenerat or f orceGenerat or i n gl obal For ceGener ators){

f orceGener ator. Appl yForce( si mObj ect ) ;}

}}

Next,wefindtheaccelerationbydividingtheresultantforcebythemassoftheobject.Wethenask

theassignedintegratortointegratetheacceleration.

( out si de Updat e( ) i n Si mul at i on. cs)

Vect or3 accel er at i on;


f or each ( Si mObj ect si mObj ect i n si mObj ect s){


/ / f i nd accel erat i onaccel er at i on = si mObj ect . Resul t ant For ce / si mObj ect . Mass;

/ / i ntegratei nt egr at or . I nt egr at e( accel er at i on, si mObj ect) ;

}}

Nowwe

update

the

object

by

asking

the

Si mObj ect

to

update

itself.

27


31/81


/ / updat e obj ectf or each ( Si mObj ect si mObj ect i n si mObj ect s){

si mObj ect . Updat e( gameTi me) ;}

RecallthattheSi mModel classimplementedUpdat ebyassigningthepositionoftheGameModel tothe

Cur r Posi t i onproperty.Thisessentiallyupdatesthepositionofthemodelthatweseeonthescreen

accordingtothenewlycalculatedCur r Posi t i on.

Finally,weresettheforcesactingontheSi mObj ect ssothattherearenoforcesintheaccumulative

Resul t ant For cepropertyatthestartofthenexttimestep.


/ / r eset f or ces on s i m obj ectsf or each ( Si mObj ect si mObj ect i n si mObj ect s){


si mObj ect . Reset For ces( ) ;}

}

1.6UsingTheNewClassesToCreateASpringSimulationNowletuscreatethesimulationinthemaingameclassbyusingalltheclassesthatwehavecreated.If

you take a look at the Game1.cs provided in the project files, it is pretty much the same as the

templateversionprovidedautomaticallybyXNAGameStudiowhenyoucreateanewXNAproject.The

only difference is that all the unnecessary comments have been removed, and that the necessary

componentshavebeenaddedin(refertothechapteronBaseCodesformoredetails).

We

shall

first

create

a

blank

method

for

initializing

the

scene:

28


32/81

( cl ass members i n Game1. cs, addi t i on of code i n bol d)


I ni t Spr i ngScene( ) ;base. I ni t i al i ze( ) ;

}

pr i vat e voi d I ni t Spr i ngScene( ){

}

Next,wewillinitializethesimulationinthenewlycreatedmethod.Wefirstloadinacubeandasphere.

Theywill serveasmarkers to indicatebothendsof thesprings.Thesemodelsare loadedusing the

Model Component fromourbasecodes.

( at t he t op of Game1. cs)


( out si de I ni t Spr i ngScene( ) i n Game1. cs)

Si mModel st at i onaryCubeSi mObj , movi ngSpher eSi mObj ;

( i nsi de I ni t Spr i ngScene( ) i n Game1. cs)

/ / l oad i n a cube and a sphere

GameModel st at i onar yCube = model Component . LoadGameModel ( @"cube" ) ;st at i onaryCube. Transl ateY = 5;GameModel movi ngSphere = model Component . LoadGameModel ( @"sphere") ;

Wethencreateaspringsimulation:


usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Si mul at i ons;

( out si de I ni t Spr i ngScene( ) i n Game1. cs)

Si mul at i on spr i ngSi m;


/ / creat e a spr i ng s i mspr i ngSi m = new Si mul at i on( t hi s) ;

29


33/81

Next,wepackthecubeandsphereintoSi mObj ect sandaddthemintothespringsimulation.Thecube

will be controlled by us, so it will be marked as passive. The sphere on the other hand is to be

controlledbythesimulation,soithastobemarkedasactive.Themassofthesphereissetto1.0after

sometrialanderror,whilethemassofthecubeissettosomearbitruaryvalue(1000.0inthiscase)but

thevaluedoesnotmattersinceitisnotusedinthecalculations.


/ / cr eat e si m obj ect s f r om t he l oaded model sf l oat spher eMass = 1. 0f ;movi ngSphereSi mObj = new Si mModel ( movi ngSpher e, spher eMass,

Si mObj ect Type. ACTI VE) ;spr i ngSi m. AddSi mObj ect ( movi ngSpher eSi mObj ) ;st at i onaryCubeSi mObj = new Si mModel ( st at i onar yCube, 1000. 0f ,

Si mObj ect Type. PASSI VE) ;spr i ngSi m. AddSi mObj ect ( st at i onaryCubeSi mObj ) ;

WethencreateaspringbetweenthetwoSi mObj ect screated:


/ / att ach a spr i ng bet ween t he si m obj ect sf l oat st i f f ness = 8. 0f ;f l oat dampi ng = 0. 1f ;spr i ngSi m. AddSpr i ng( st i f f ness, dampi ng, st at i onaryCubeSi mObj ,

movi ngSpher eSi mObj ) ;

Next,weaddglobalforcestothesimulationsystem.Wewillbeusingthegravitationalaccelerationof

Earth(9.81downwards)forthegravity.Forthedragcoefficientvalue,avalueof0.5wasdeemedtobe

appropriateaftersometrialanderror.


usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. For ceGener at or s;


/ / add i n a gl obal f or ce gener at or : gr avi t yGr avi t y gr avi t y = new Gr avi t y( new Vect or3(0, - 9. 81f , 0) ) ;spri ngSi m. AddGl obal For ceGener ator( gr avi t y) ;

30


34/81

/ / add i n a gl obal f or ce gener at or : ai rf l oat dr agCoef f i ci ent = 0. 5f ;Medi um ai r = new Medi um( dr agCoef f i ci ent ) ;spr i ngSi m. AddGl obal For ceGener at or( ai r ) ;

Nowwe

create

an

I nt egr at orand

assign

it

to

the

simulation.


usi ng Skeel Sof t BodyPhysi csTut or i al . Sof t Body. I nt egr at or s;


/ / creat e an i nt egr at or and assi gn i t t o t he si m For war dEul er I nt egr at or i nt egr at or = new For war dEul er I nt egr ator ( t hi s) ;

spr i ngSi m. I nt egr at or = i nt egr at or ;

Inordertovisualizethespring,weshallalsodrawalinefromthecubetothespherethatrepresents

thespring.AlinehasalreadybeencreatedwhenweincludeLi ne3DComponent intogame. Component s,

soallwehavetodonowistosetthepositionsofthetwoends,andalsothecoloroftheline.


/ / i ni t t he l i ne f r om cube t o spher e that r epr esent s t he spr i ngl i ne3DComponent . St art Posi t i on = st at i onaryCube. Transl ate;l i ne3DComponent . EndPosi t i on = movi ngSpher e. Transl at e;l i ne3DComponent . Col or = Col or. Whi t e;

Now,weneedtoupdatethesimulationandthelineintheUpdat emethod.Allwehavetodoistoask

thesimulationtoupdateitself,andthenmovethetwoendsofthelinetothenewpositionscalculated

forthesphereandcubesimulationobjects.

31


35/81

( i nsi de Updat e( ) i n Game1. cs, addi t i on of code i n bol d)


/ / updat e t he si mspr i ngSi m. Updat e( gameTi me) ;

/ / updat e l i nel i ne3DComponent . St ar t Posi t i on = st at i onar yCubeSi mObj . Cur r Posi t i on;l i ne3DComponent . EndPosi t i on = movi ngSpher eSi mObj . Curr Posi t i on;


Ifyoucompileandexecutetheprogramnow,youshouldhaveastationarycubeandamovingsphere

thatisattachedviaaspring,whichmeansthatthesimulationthatwehavecodedworks!

Letusnowdothelastbitsofcodingtoallowtheusertocontrolthestationarycube.

( cl ass met hods i n Game1. cs, addi t i on of code i n bol d)


/ / pol l f or i nputHandl eI nput ( gameTi me) ;

/ / updat e t he si mspr i ngSi m. Updat e( gameTi me) ;

/ / updat e l i nel i ne3DComponent . St art Posi t i on = st at i onaryCubeSi mObj . Cur r Posi t i on;l i ne3DComponent . EndPosi t i on = movi ngSpher eSi mObj . Curr Posi t i on;


pri vat e voi d Handl eI nput ( GameTi me gameTi me){

i f ( i nput Component . I sKeyHel dDown(Keys. Ri ght ) ){

st at i onaryCubeSi mObj . Cur r Posi t i onX += 0. 1f ;}

i f ( i nput Component . I sKeyHel dDown(Keys. Le f t ) ){st at i onar yCubeSi mObj . Cur r Posi t i onX - = 0. 1f ;

}i f ( i nput Component . I sKeyHel dDown(Keys. Up) ){

st at i onar yCubeSi mObj . Cur r Posi t i onY += 0. 1f ;}

32


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i f ( i nputComponent . I sKeyHel dDown(Keys. Down) ){

st at i onar yCubeSi mObj . Cur r Posi t i onY - = 0. 1f ;}

}

Youshouldnowhaveaspringsimulationthatcanbecontrolledusingthekeyboard,just likeVideo4

thatyousawinthebeginningofthischapter.

Inthischapter,wehavelookedathowaphysicssimulationshouldbedoneingames.Wethencreated

fourmaingroupsofclassesthathelpuscreatethesesimulationsinanobjectorientedmanner.Finally,

wemadeuseofalltheseclassestocreateasimplespringsimulation.

Thischapterisquitelongbecauseitcoversalotofbasicsforsoftbodysimulations.Westillhavethree

morechapterstogobutdonotworry,mostoftheworkhasalreadybeendoneinthischapter.Inthe

nextchapter,

we

shall

start

to

implement

acloth

simulation,

and

it

will

rely

largely

on

the

codes

that

wehavecreatedsofar.

GetthesourcefilesfortheendofthischapterfromtheSkeelSoftBodyPhysicsTutorialChapter1ENDfolder

33


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Chapter2 - ClothSimulation

GetthesourcefilesforthestartofthischapterfromtheSkeelSoftBodyPhysicsTutorialChapter2BEGINfolder

Inthischapter,wewillbecreatingasimpleclothsimulation.Aclothisoneofthebasicsoftbodiesthat

issimple

to

create

and

yet

produces

pleasant

results.

Softbodies,ingeneral,ismostlyabouthowyouconnectthespringsbetweenvertices.Eachvertexof

themodel isabletomove independentlyfromothers,allowingthemodeltodeformandchange its

overallshape.Whetherthemodeldeformslikeasoftbodythatyouexpectverymuchdependsonhow

youconnect thespringsand theattributesof thesprings.Now thatweknowhow tocreatesprings

from thepreviouschapter,wewill learnhow toconnect the springs for theplanegeometry in this

chaptersothataclothsimulationcanbecreated.

Bytheendofthischapter,youshouldbeabletocreateacloththatlookslikethis:

Video5:Aclothsimulationcreatedusingsimplespringsandconstraints[Clickonimageabovetowatchthevideo]

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2.1ConstraintsBeforewegetintocreatingtheclothitself,letusfirsttakealookatconstraintsandwhyweneedthem.

Constraints

are

basically

conditions

that

the

simulation

system

must

try

to

satisfy.

Some

examples

of

constraintsare lengthconstraintswhere twoobjects try tomaintain theirdistanceapart,andpoint

constraintswhereanobjecttriestosticktoacertainpointinspace.

Inordertounderstandwhylengthconstraintsforclothareso important, Ihavecreatedaversionof

the cloth thatdoes not use them.Notice how thewhole cloth looks too elastic due to the highly

springybehavior,especiallyatthepinnedcorners.Itisalsoquiteunstableattimes.

Video6:Aclothsimulationwithoutlengthconstraints.Theclothtendstobetooelasticandquiteunstableattimes.


Whatwe

really

want

for

acloth

simulation

is

to

have

length

constraints

between

the

vertices

on

top

of

thesprings.Thiswillhelpmaintainthedistancesbetweentheverticesandgiveamuchmorerealistic

behavior for thecloth simulation.Wewillalsowant tohavepointconstraintsat thecornersof the

clothsothatthecornerscanbepinnedinspacefortheusertocontrol.

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Thewaywe solve theconstraintswillbeby relaxation,averypracticalmethod thatworkswell for

games.Wewilliteratennumberoftimesandateachiteration,wetrytogeteachconstrainttosatisfy

thegoal.Forexample, forapointconstraint,wecouldperhapsdoa10iteration loopandatevery

iteration,wemovetheobjecttowardsthegoalpositionasspecifiedbythepointconstraint.

Why

then

do

we

want

to

iterate

n

times

and

get

the

constraint

to

move

towards

the

goal

repeatedly?

Thereason isthattheremightbemanydifferentconstraintstobesatisfiedandbysatisfyingoneof

them, the other constraints might be violated. Imagine a simulation with two connected length

constraints.Duringeachiteration,thefirstconstraintmovestowardsthegoallengthbutbydoingso,

thegoallengthofthesecondconstraintmighthavebeenincreased.Thesecondconstraintthenmoves

towardsitsgoallengthandinturnmighthavecausedthegoallengthofthefirstconstrainttoincrease.

Thisseemsliketheyarecounteringtheeffectofeachother,butattheendofeachiteration,bothof

themwouldhavemovedtowardstheirgoallengthstosomeextent(convergenceisguaranteedunder

therightconditions).Thisisthereasonwhywehavetoiteratethroughnnumberoftimessothatwe

are slowly relaxing the fight between the constraints and theywill slowly converge to their goal

lengthsrespectively.

Thenumberofiterationstoperformdependsonhowmuchprocessingpoweryouhaveforyourgame.

Thehigherthe iterations,thebettertheconstraintswillbesatisfied,butthemoreexpensive it isto

calculate.Amoderatenumberofabout10to20oftenworkswelltogivesatisfactoryresult,sothereis

usually no need to increase this number to high values like 100. In fact, in some cases,just one

iterationmightbeenough.Itissuggestedthatyoureducethisnumbertoasuitablevaluewhichgives

visuallyacceptableresults.

2.1.1ConstraintInterface

CreateafoldercalledConstraintsintheSoftBodyfolder,andaddanewclassfilenamedConstraint.cs.

The Const r ai nt classwillbasicallybean interface thatmakes it compulsory forderived classes to

implementamethodcalledSat i sf yConst r ai nt .

( Const r ai nt . cs)

namespace Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Const r ai nt s{

publ i c i nt erf ace Const r ai nt{

voi d Sat i sf yConstr ai nt ( ) ;}

}

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2.1.2LengthConstraint

Asmentionedearlier, lengthconstraintsarebasicallyconstraints that tries tomaintain thedistance

betweentwoobjects.

IntheConstraintsfolder,addanewclassfilenamedLengthConstraint.cs.ALengt hConst r ai nt class

basicallytakes

in

two

Si mObj ect sand

the

expected

distance

between

them.

( Lengt hConst r ai nt . cs)



publ i c seal ed cl ass LengthConst r ai nt : Const r ai nt{

pr i vat e f l oat l engt h;pr i vat e Si mObj ect si mObj 1;pr i vat e Si mObj ect si mObj 2;

publ i c f l oat Length{

get { return l engt h; }set { l engt h = val ue; }

}

/ / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

publ i c Lengt hConst r ai nt ( f l oat l engt h, Si mObj ect si mObj 1, Si mObj ect si mObj 2){

t hi s. l engt h = l engt h;t hi s. si mObj 1 = si mObj 1;t hi s. si mObj 2 = si mObj 2;

}}

}

Next, we have to implement the Sat i sf yConst r ai nt method. Recall that we have to satisfy the

constraint at each iteration. For a length constraint, this canbedonebymovingboth ends of the

constraintbyhalfthedifferenceofthecurrentlengthandthegoallength,asshowninFig.2.

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Figure2:Twoendsofalengthconstraintmovingtowardsthegoallengthinasingleiteration

The implementationofthat isgivenbelow.Basicallywecalculate thedirectionandcurrentdistance

between the twoSi mObj ect s.Thenwenormalize thedirectionvector,andmovebothSi mObj ect s

halfthedifferencedistanceintheconvergingdirection.

( cl ass met hod i n Lengt hConst r ai nt . cs)

Vect or3 di r ecti on;f l oat cur r ent Lengt h;Vect or3 moveVect or ;publ i c voi d Sat i sf yConst r ai nt ( ){

/ / cal cul at e di rect i ondi r ect i on = si mObj 2. Cur r Posi t i on - si mObj 1. Cur r Posi t i on;

/ / cal cul at e cur r ent l engt hcur r ent Lengt h = di r ect i on. Lengt h( ) ;

/ / check f or zer o vect ori f ( di r ect i on != Vect or3. Zer o){

/ / nor mal i ze di r ect i on vect ordi r ecti on. Nor mal i ze( ) ;

/ / move t o goal posi t i onsmoveVect or = 0. 5f * ( cur r ent Lengt h - l engt h) * di r ect i on;si mObj 1. Curr Posi t i on += moveVect or ;si mObj 2. Curr Posi t i on += - moveVect or ;

}}

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2.1.3PointConstraint

Anotheruseful typeofconstraint isapointconstraint.Apointconstraint tries topinanobject toa

certainpointinspace.Thisisusefulforconstrainingthetoptwocornersofourclothlateron.Byusing

point constraints,we are also able to control the twopinned cornersby changing their respective

constraintposition.

IntheConstraintsfolder,createanewclassfileandcallitPointConstraint.cs.

( Poi nt Const r ai nt . cs)



publ i c seal ed cl ass Poi nt Const r ai nt : Const r ai nt{

pr i vat e Vect or3 poi nt ;pr i vat e Si mObj ect si mObj ect ;

publ i c Vect or3 Poi nt{

get { return poi nt ; }set { poi nt = val ue; }

}

publ i c f l oat Poi nt X{

get { return poi nt . X; }

set { poi nt . X = val ue; }}

publ i c f l oat Poi nt Y{

get { return poi nt . Y; }set { poi nt . Y = val ue; }

}

publ i c f l oat Poi nt Z{

get { return poi nt . Z; }set { poi nt . Z = val ue; }

}

publ i c Si mObj ect Si mModel{

get { return si mObj ect ; }set { si mObj ect = val ue; }

}

/ / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

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publ i c Poi nt Const r ai nt ( Vect or3 poi nt , Si mObj ect si mObj ect ){

t hi s. poi nt = poi nt ;t hi s. si mObj ect = si mObj ect ;

}}

}

NowwehavetoimplementtheSat i sf yConst r ai nt method.Tosatisfyapointconstraint,allwehave

todoismovetheobjecttothegoalposition.

( cl ass met hod i n Poi nt Const r ai nt . cs)

publ i c voi d Sat i sf yConst r ai nt ( ){

/ / move t o goal posi t i onsi mObj ect . Cur r Posi t i on = poi nt ;}

2.1.4ProcessingTheConstraintsInTheSimulationClass

NowthatwehaveboththeLengt hConst r ai nt andPoi nt Const r ai nt classescreated,weneedtolet

theSi mul at i onclassprocessthem.

First,weaddinsomevariablesinSimulation.cstokeeptrackoftheConst r ai nt s.Thisconsistsofalist

of Const r ai nt sanda constr ai nt I t er at i ons variable thatdetermineshowmany times to iterate

throughthe

Const r ai nt s

per

time

step.

( at t he t op of Si mul at i on. cs)

usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Const r ai nt s;

( cl ass member s i n Si mul at i on. cs)

pr ot ect ed Li st const r ai nt s = new Li s t( ) ;pr otect ed i nt constr ai nt I t er at i ons;

publ i c Li s t Const r ai nt s{

get { return constr ai nt s; }set { const r ai nt s = val ue; }

}

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publ i c i nt Const r ai nt I t er at i ons{

get { return constr ai nt I t er at i ons; }set { const r ai nt I t er at i ons = val ue; }

}

Thenwe

have

to

iterate

through

the

constraints

to

relax

them,

as

described

in

the

beginning

of

Section

2.1.Thisrelaxationiterationhastobedoneaftertheintegratorhassolvedforthenewobjectpositions,

butbeforeupdatingtheobjects.

( cl ass met hod i n Si mul at i on. cs, addi t i on of code i n bol d)

publ i c vi r t ual voi d Updat e( GameTi me gameTi me){

/ / sum al l l ocal f orcesf oreach ( Spr i ng spr i ng i n spr i ngs)

{spr i ng. Appl yFor ce( nul l ) ; / / no need t o speci f y any si mObj

}

/ / sum al l gl obal f or ces acti ng on t he obj ectsf oreach ( Si mObj ect si mObj ect i n si mObj ect s){


f or each ( ForceGenerat or f orceGenerat or i n gl obal For ceGener ators){

f orceGener ator. Appl yForce( si mObj ect ) ;}

}}

f oreach ( Si mObj ect si mObj ect i n si mObj ect s){


/ / f i nd accel erat i onaccel er at i on = si mObj ect . Resul t ant For ce / si mObj ect . Mass;

/ / i ntegratei nt egr at or . I nt egr at e( accel er at i on, si mObj ect) ;

}}

/ / sat i s fy constr ai nt sf or ( i nt i = 0; i < const r ai nt I t er at i ons; i ++){

f oreach ( Const r ai nt const r ai nt i n const r ai nt s){

constr ai nt . Sat i sf yConstr ai nt ( ) ;}

}

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/ / updat e obj ectf or each ( Si mObj ect si mObj ect i n si mObj ect s){

si mObj ect . Updat e( gameTi me) ;}

/ / r eset f or ces on s i m obj ectsf oreach ( Si mObj ect si mObj ect i n si mObj ect s){


si mObj ect . Reset For ces( ) ;}

}}

2.2Verlet(NoVelocity)IntegratorItistimetointroduceanotherintegratorcalledtheVerletintegrator.Itisknowntobefastandstable

(secondorder,ascomparedtofirstorderforForwardEuler),andthushasbeenapopularchoicefor

games.There are severalvariantsofVerlet integrators available, suchasVelocityVerlet and also a

multistepmethodcalledLeapFrogVerlet.However,theVerletintegratorthatweareinterestedinisa

versionthattotallyomitstheobjectsvelocityinitscalculations.

Intheprevioussection,wehavecoveredhowtosolveconstraintsbyrelaxation.Ifyouhaverealized,in

ordertosatisfytheconstraints,wesimplymovedtheobjectstowardstheirgoalpositionswithoutany

consideration for theirvelocities!This iswheretheVerlet integratorcomes in.Byexplicitlyomitting

the velocities in the calculations,we canperform the simple shiftingofpositionsduring relaxation

withoutworryingaboutthevelocities,andwewillstillbeabletosolveforthenewpositionsateach

timestep.

TheequationfortheVerletintegrationisasfollows:

x(t+ t)=2x(t)+x(t t)+a(t)(t)2

[8]

Itbasicallyusesthepositioninthecurrentandprevioustimestepstodeterminethepositionforthe

nexttimestep.

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TheaboveVerletintegrationequationcanbemodifiedtosimulatedragforcesinthemedium,whered

isadragcoefficientbetween0and1inclusive.

x(t+ t)=(2d)x(t)+(1d)x(t t)+a(t)(t)2

[9]

ByusingEqn.9,wecan leaveouttheMedi umforcegenerator inoursimulation lateron,sincethed

variableintheequationaboveisagoodsubstituteforthat.Wewillbeusingthismodifiedequationin

oursimulationfromnowon.

Letusnow start implementing thisVerlet integrator. Inorder todifferentiate thisVerlet integrator

from theothervariants,we shallname thisclass Ver l et NoVel oci t yI nt egr at or . In the Integrators

folder,createanewclassfilenamedVerletNoVelocityIntegrator.cs.

( Ver l et NoVel oci t yI nt egr at or . cs)

usi ng Syst em;usi ng Mi cr osof t . Xna. Framework;usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Si mObj ect s;


publ i c seal ed cl ass Ver l et NoVel oci t yI nt egr at or : I nt egr at or{

pr i vat e f l oat drag;

publ i c f l oat Dr ag{

get { return dr ag; }set{

i f ( val ue < 0 | | val ue > 1){

throw new Ar gument Except i on( "Ai r r esi st ance must be between 0and 1" ) ;

}drag = val ue;

}

}

publ i c Ver l et NoVel oci t yI nt egr at or ( Game game) : t hi s( game, 0. 005f ) { }

publ i c Ver l et NoVel oci t yI nt egr at or ( Game game, f l oat drag): base( game)

{t hi s. drag = drag;

}

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}}

Thisclassbasicallytakesinadragvalueintheconstructor.Itwillbesetto0.005bydefaultifnovalue

is

provided.

WenowneedtoimplementthecompulsoryI nt egr at emethod,usingEqn.9.

( cl ass met hod i n Ver l et NoVel oci t yI nt egr at or . cs)

Vect or3 newPosi t i on;publ i c over r i de voi d I nt egr at e( Vect or3 accel er at i on, Si mObj ect si mObj ect ){

newPosi t i on = ( 2 - dr ag) * si mObj ect . Cur r Posi t i on- ( 1 - dr ag) * si mObj ect . Pr evPosi t i on

+ accel er at i on * f i xedTi meSt ep * f i xedTi meSt ep;

si mObj ect . Pr evPosi t i on = si mObj ect . Cur r Posi t i on;si mObj ect . Cur r Posi t i on = newPosi t i on;

}

2.3SimulationVertexRecallthatinthepreviouschapter,wepackedamodelintoasimulationobjectcalledSi mModel sothat

thesimulation

attributes

such

as

position,

velocity

and

mass

are

included.

For

acloth,

we

are

interested inmoving itsvertices,meaningthatthesimulationobjectsareactuallytheverticesofthe

clothitself!Wewillthusneedtoimplementanewsimulationclassforthevertices.

IntheSimObjectsfolder,createanewclassfilenamedSimVertex.cs.

( Si mVert ex. cs)

usi ng Mi cr osof t . Xna. Framework;usi ng Skeel Sof t BodyPhysi csTut or i al . Pr i mi t i ves;


publ i c seal ed cl ass Si mVer t ex : Si mObj ect{

pr i vat e i nt ver t exI d;pr i vat eText uredPri mi t i ve pr i mi t i ve;

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publ i c i nt Ver t exI d{

get { return ver t exI d; }set { ver t exI d = val ue; }

}

/ / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

publ i c Si mVer t ex( f l oat mass, Si mObj ect Type si mObj ect Type, i nt ver t exI d,Text uredPr i mi t i ve pr i mi t i ve)

: base( mass, si mObj ect Type){

t hi s. ver t exI d = ver t exI d;t hi s. pr i mi t i ve = pr i mi t i ve;t hi s. cur r Posi t i on = pr i mi t i ve. Get Ver t exPosi t i on( ver t exI d) ;t hi s. pr evPosi t i on = cur r Posi t i on;

}}

}

TheSi mVert exclasstakesinthemassofthevertex,theSi mObj ect Type(activeorpassive),thevertex

idandtheprimitive.Allthese informationwillallowustoretrieveandsetthevertexpositionswhen

weneedto.TheothernecessaryinformationwillcomefromthebaseclassSi mObj ect .

Wewillhave to implement the Updat emethodaswell.Allwehave todo is tojust set the vertex

positiontothenewlycalculatedposition.

( cl ass method i n Si mVert ex. cs)


pr i mi t i ve. Set Ver t exPosi t i on( t hi s. ver t exI d, t hi s. curr Pos i t i on) ;}

2.4SoftBodyandClothSimulationClassesWewillnowproceedonbycreatingclassesthatrepresentasoftbodyandacloth.Aclothisessentially

a

soft

body,

so

the

Cl ot hSi mclass

that

we

will

be

creating

will

inherit

from

the

Sof t BodySi mclass.

Let

us firststartoffbycreatingtheSof t BodySi mclass. IntheSimulations folder,createanewclass file

namedSoftBodySim.cs.

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( Sof t BodySi m. cs)



publ i c c l ass Sof t BodySi m : Si mul at i on{pr ot ect ed Si mVer t ex[ ] si mVer t i ces;

publ i c Si mVer t ex[ ] Si mVer t i ces{

get { return si mVer t i ces; }set { si mVer t i ces = val ue; }

}

/ / - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

publ i c Sof t BodySi m( Game game): base( game) { }

}}

TheSof t BodySi mclassbasicallyinheritsfromtheSi mul at i onclass,andstoresalistofSi mVer t ex.

Next,forourCl ot hSi mclass,createanewclassfilenamedClothSim.csintheSimulationsfolder.

( Cl ot hSi m. cs)

usi ng Mi cr osof t . Xna. Framework;usi ng Mi cr osof t . Xna. Framework. Gr aphi cs;usi ng Skeel Sof t BodyPhysi csTut or i al . Pr i mi t i ves;usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Const r ai nt s;usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Si mObj ect s;


publ i c seal ed cl ass Cl ot hSi m : Sof t BodySi m{

pr i vat eText uredPl ane cl ot hPl ane;

publ i c Cl ot hSi m( Game game, Text uredPl ane cl ot hPl ane, f l oat cl ot hMass,f l oat str uct St i f f ness, f l oat st r uct Dampi ng,f l oat shear St i f f ness, f l oat shearDampi ng,f l oat bendSt i f f ness, f l oat bendDampi ng)

: base( game){

t hi s. cl othPl ane = cl ot hPl ane;

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/ / creat e s i m dat aCr eat eSi mVer t i ces( cl ot hPl ane, cl ot hMass);

/ / connect spr i ngs and add const r ai nt sConnect Spr i ngs( st r uct St i f f ness, st r uct Dampi ng,

shear St i f f ness, shearDampi ng,bendSt i f f ness, bendDampi ng) ;

}}}

TheCl ot hSi mclasstakesinaText uredPl aneastheclothgeometry,aswellastheattributesforthe

differenttypesofspringsneededto linkuptheclothvertices. Italsocreatesthesimulationvertices,

connects thespringsandadds lengthconstraintsbetween thevertices.The twomethodcalls inthe

constructorhavenotbeenimplementedyet,andwillbediscussednow.

TheimplementationforCr eat eSi mVert i cesisasfollows:

( cl ass met hod i n Cl othSi m. cs)

pr i vat e voi d Cr eat eSi mVer t i ces(Text uredPl ane cl ot hPl ane, f l oat cl othMass){

i nt numVert i ces = cl ot hPl ane. NumVert i ces;f l oat vert exMass = cl ot hMass / numVer t i ces;si mVer t i ces = new Si mVer t ex[ numVer t i ces] ;f or ( i nt i = 0; i < numVert i ces; i ++){

si mVer t i ces[i ] = new Si mVer t ex( ver t exMass, Si mObj ect Type. ACTI VE, i ,cl ot hPl ane) ;

t hi s. AddSi mObj ect ( si mVer t i ces[ i ] ) ;}

}

ItbasicallycreatesaSi mVer t exforeachvertexoftheplane,andsetsittoactivesothattheywillbe

processedbythesimulation.

BeforewediscussabouttheConnect Spr i ngsmethod,weneedtotalkabouthowthespringsshould

beconnectedfortheclothgeometry.Thereareessentiallythreekindsofspringsthatareneededfora

goodcloth

simulation:

1. Structuralsprings(Fig.3):Theseareplacedalongeachquadedge.Theyarethebasicspringswhichmaintaintheinterdistancebetweentheverticesinthehorizontalandverticaldirections.

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Figure3:Structuralspringsonaplane

2. Shearsprings(Fig.4):Theseareplacedalongthediagonalsofeachquad.Theyhelptomaintainthe square shape of each quadon the geometry, preventing them from collapsing into flat

diamondshapes.

Figure4:Shearspringsonaplane

3. Bend springs (Fig. 5 and 6): These are placed horizontally and vertically as well, just likestructuralsprings.However,theyareplacedbetweenverticesthataretwoverticesapart.They

helptocontrolhowmuchtheclothcanbend.Ahighstiffnessvalueforthesespringswillcreate

astifferlook(e.g.leather)whilealowstiffnessvaluewillcreateasofterbehavior(e.g.silk).

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Figure5:Horizontalbendspringsonaplane

Figure6:Verticalbendspringsonaplane

Thesevideosillustratehowtheclothwillbehavewithoutsomeofthesesprings:

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Video7:Clothsimulationwithonlystructuralsprings(noshearandbendsprings).Theclothisunabletomaintainagoodshapewhendrapingdownwards.


Video8:Clothsimulationwithstructuralandshearsprings(nobendsprings).Theclothholdsitsshapebetterbutisunabletobendrealisticallyandpenetratesitselfeasily.


50
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Youcanprobablygetawaywithnotusingthebendsprings incertaincases,suchasapieceofcloth

that isflutteringwiththewind,as longas it isvisuallyplausible.Thiscanactuallyhelptosavesome

springandconstraintcalculations.Wewill,however,beusingthebendspringsinoursimulation.

With this new knowledge of how to connect the springs, we can start to implement the

Connect Spr i ngs

method:


pr i vat e voi d Connect Spr i ngs( f l oat str uctSt i f f ness, f l oat st r uct Dampi ng,f l oat shear St i f f ness, f l oat shearDampi ng,f l oat bendSt i f f ness, f l oat bendDampi ng)

{f or ( i nt x = 0; x < cl ot hPl ane. Lengt hSegment s; x++){

f or ( i nt y = 0; y


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t hi s. AddSpr i ng( shearSt i f f ness, shearDampi ng,si mVer t i ces[ ver t exAI d] , si mVer t i ces[ ver t exBI d] ) ;

f l oat l engt h = ( cl othPl ane. Get Ver t exPosi t i on( ver t exAI d) -cl ot hPl ane. Get Ver t exPosi t i on( ver t exBI d) ) . Lengt h( ) ;

t hi s. Const r ai nt s. Add( new Lengt hConst r ai nt ( l engt h,si mVer t i ces[ ver t exAI d] , si mVer t i ces[ ver t exBI d] ) ) ;

/ / shear spr i ng: di agonal ( \ )ver t exAI d = x + y * ( cl ot hPl ane. LengthSegment s + 1) ;ver t exBI d = ( x + 1) + ( y + 1) * ( cl ot hPl ane. LengthSegment s + 1) ;t hi s. AddSpr i ng( shearSt i f f ness, shearDampi ng,

si mVer t i ces[ ver t exAI d] , si mVer t i ces[ ver t exBI d] ) ;l engt h = ( cl ot hPl ane. Get Ver t exPosi t i on( ver t exAI d) -

cl ot hPl ane. Get Ver t exPosi t i on( ver t exBI d) ) . Lengt h( ) ;t hi s. Const r ai nt s. Add( new Lengt hConst r ai nt ( l engt h,

si mVer t i ces[ ver t exAI d] , si mVer t i ces[ ver t exBI d] ) ) ;}

}

f or ( i nt x = 0; x < cl ot hPl ane. Lengt hSegment s - 1; x++){

f or ( i nt y = 0; y


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That isquitea lotofcode,butmostof the codesareactually repetitive.The structural springsare

createdfirst,followedbytheshearspringsandthenthebendsprings.Lengthconstraintsarecreated

inthesameloops.

Nowthatwehavethespringsconnected,weneedtoupdatetheclothgeometryintheUpdat emethod:



/ / cal l base. Updat e( ) t o updat e the ver t ex posi t i onsbase. Updat e( gameTi me) ;

/ / r ecal cul at e the ver t ex nor mal scl othPl ane. Recal cul at eNor mal s( ) ;

/ / commi t t he ver t ex posi t i on and normal changes

cl ot hPl ane. Commi t Changes( ) ;}

Wefirstcallthebaseclasstoupdateitselffirst,whichwillcalculateallthenewpositionsofthevertices

and set the clothvertices to thosenewpositions.Since theverticesmove independently fromone

another,theirrelativepositionschangeovertimeandtheirnormalswillchangeaswell.Thus,weneed

to recalculate the normals so that the rendering is faithful to the geometry changes. Note that

Recal cul at eNormal s has already been implemented in the base class. Finally, we need to call

Commi t Changeson

the

plane

geometry

which

will

update

all

the

vertex

position

changes

in

the

video

RAM.

2.5UsingTheNewClassesToCreateAClothSimulationWearenowreadytocreatetheclothinourmainprogramfileGame1.cs.

First,weinitializethescene.

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( cl ass met hod i n Game1. cs, addi t i onal codes i n bol d)


I ni t Cl ot hScene( ) ;base. I ni t i al i ze( ) ;

}

pri vat e voi d I ni t Cl ot hScene( ){

}

Wedefinetheclothattributesandcreateaplaneastheclothgeometry.


usi ng Skeel Sof t BodyPhysi csTut or i al . Pr i mi t i ves;

( out si de I ni t Cl ot hScene( ) i n Game1. cs)

Text uredPl ane cl ot hPl ane;

( i nsi de I ni t Cl othScene( ) i n Game1. cs)

/ / c l ot h at t r i but esi nt l ength = 10;i nt wi dt h = 5;i nt l engthSegment s = 20;

i nt wi dt hSegment s = 15;

/ / l oad i n a pl anecl ot hPl ane = newText uredPl ane( t hi s, l engt h, wi dt h, l engt hSegment s,

wi dt hSegments, @"checker board") ;cl ot hPl ane. I ni t i al i ze( ) ;

IntheSolutionExplorer,noticethatthereisafoldercalledPrimitivesthatcontainsclassesthatyoucan

usetocreatetexturedplanesandcylinders.Iwillnotbeexplainingthecodes intheseclassesbutdo

notethatwiththeseclasses,youwillbeabletocreateprimitivesandalsoget/setthevertexattributes

suchas

the

position

and

normal.

If

you

are

going

to

use

your

own

primitive

classes

or

load

your

own

modelsfromfiles,youwillalsoneedtohavemethodstoget/setthevertexattributes.

We thendefine themassof thecloth,attributesof thedifferent springs for thecloth,andcreatea

Cl ot hSi minstance:

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usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Si mul at i ons;


Cl ot hSi m cl othSi m;

( i nsi de I ni t Cl ot hScene( ) i n Game1. cs)

/ / creat e a cl ot h s i m f l oat cl othMass = 2. 0f ;

f l oat st r uctSti f f ness = 2. 0f ;f l oat st r uctDampi ng = 0. 02f ;f l oat shear St i f f ness = 2. 0f ;f l oat shear Dampi ng = 0. 02f ;f l oat bendSt i f f ness = 2. 0f ;f l oat bendDampi ng = 0. 02f ;cl ot hSi m = new Cl ot hSi m( t hi s, cl ot hPl ane, cl ot hMass, st r uctSti f f ness,

st r uct Dampi ng, shear St i f f ness, shearDampi ng,bendSt i f f ness, bendDampi ng) ;

cl ot hSi m. Const r ai nt I t er at i ons = 10;

Next,wecreate theglobal forcegenerators.Forourcloth simulation, thiswillonlybegravity.Note

thatwewillnotbecreatingaMedi umforcegeneratorsincethedrag forceswillbesimulated in the

Verletintegrator.


usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. For ceGener at or s;


/ / add i n a gl obal f or ceGener at or s: gr avi t yGr avi t y gr avi t y = new Gr avi t y( new Vect or3(0, - 9. 81f , 0) ) ;cl ot hSi m. AddGl obal For ceGener at or( gr avi t y) ;

Wewillconstrainthetopleftandtoprightcornersoftheplaneinordertocontrolthemlateron.This

isdonebyusingPoi nt Const r ai nt s.


usi ng Skeel Sof t BodyPhysi csTut ori al . Sof t Body. Const r ai nt s;

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Poi nt Const r ai nt t opLef t Cor ner , t opRi ght Cor ner ;


/ / const r ai n t he t wo t op cor ner s of t he cl ot h so t hat we can cont r ol i t

t opLef t Cor ner = newPoi nt Const r ai nt ( cl ot hSi m. Si mVer t i ces[ 0] . Cur r Posi t i on, cl ot hSi m. Si mVer t i ces[ 0] ) ;cl ot hSi m. Const r ai nt s. Add( t opLef t Cor ner ) ;t opRi ght Corner = new

Poi nt Const r ai nt ( cl othSi m. Si mVer t i ces[ l engt hSegment s] . Cur r Posi t i on,cl othSi m. Si mVer t i ces[ l engt hSegment s] ) ;

cl othSi m. Const r ai nt s. Add( t opRi ght Cor ner ) ;

WethencreateaVerletintegrator,withanassociateddragvalue.


usi ng Skeel Sof t BodyPhysi csTut or i al . Sof t Body. I nt egr at or s;


/ / creat e an i nt egr at or and assi gn i t t o t he si m f l oat drag = 0. 01f ;

I nt egr at or i nt egr at or = new Ver l et NoVel oci t yI nt egr at or ( t hi s, dr ag) ;cl ot hSi m. I nt egr at or = i nt egr at or ;

Thereareafewmorestepsregardingourclothplanebeforewecanexecutetheprogram.Weneedto

callitsLoadCont ent methodsothatitcanloadthenecessarycontents.

( cl ass met hod, addi t i onal code i n bol d)

pr ot ect ed over r i de voi d LoadCont en

IntroductionToSoftBodyPhysics-Skeel

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