Physics-basedSimula1on

•  simple(independentpar1cles),orcomplex(robustcolliding,stacking,sliding3Drigidbodies)

•  manymanysimulators!–  PhysX(Unity,Unreal),Bullet,OpenDynamicsEngine,MuJoCo,Havok,Box2D,Chipmunk,OpenSim,RBDL,Simulink(MATLAB),ADAMS,SD/FAST,DART,Vortex,SOFA,Avatar,ProjectChrono,Cannon.js,…

–  manycourseprojects,theses,abandon-ware

Resources•  hUps://processing.org/examples/see“Simulate”;2Dpar1clesystems•  Non-convexrigidbodieswithstacking3Dcollisionprocessingandstacking

hUp://www.cs.ubc.ca/~rbridson/docs/rigid_bodies.pdf•  Physically-basedModeling,coursenotes,SIGGRAPH2001,Baraff&Witkin

hUp://www.pixar.com/companyinfo/research/pbm2001/•  DougJamesCS5643coursenotes

hUp://www.cs.cornell.edu/courses/cs5643/2015sp/•  RigidBodyDynamics,ChrisHeckerhUp://chrishecker.com/Rigid_Body_Dynamics•  Videogamephysicstutorial

hUps://www.toptal.com/game/video-game-physics-part-i-an-introduc1on-to-rigid-body-dynamics•  Box2DjavascriptlivedemoshUp://heikobehrens.net/misc/box2d.js/examples/•  RigidbodycollisionsjavascriptdemohUps://www.myphysicslab.com/engine2D/collision-en.html•  RigidBodyCollisionReponse,MichaelManzke,courseslides

hUps://www.scss.tcd.ie/Michael.Manzke/CS7057/cs7057-1516-09-CollisionResponse-mm.pdf•  Interac1vesimula1onofrigidbodydynamicsincomputergraphics,CGF2014

hUp://onlinelibrary.wiley.com/doi/10.1111/cgf.12272/abstract•  AMathema1calIntroduc1ontoRobo1cManipula1on(textbook)

http://www.cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf•  Par1cle-basedFluidSimula1onforInterac1veApplica1ons,SCA2003,PDF•  StableFluids,JosStam,SIGGRAPH1999.interac1vedemo:

hUps://29a.ch/2012/12/16/webgl-fluid-simula1on

Physically-basedSimula1on–movementgovernedbyforces

•  par1clesystems–  fire,water,smoke,(fluids,cloth,hair)–  heuris1cs-basedormoreprincipled

•  rigid-bodysimula1on–  blocks,robots,humans

•  con1nuumsystems–  deformablesolids–  fluids,cloth,hair

•  groupmovement–  flocks,crowds

OrdinaryDifferen1alEqua1onsdX(t)dt

= f (X(t), t)Given X0 = X(t0 )

X(t)Compute t > t0for

Simula1onasapaththroughstate-space,drivenbyavectorfield

PendulumSpring

Simula1onBasics

•  simula1onloop– Equa1onsofMo1on

solvefordX/dt=f(X,forces,torques)– numericalintegra1on

updateposi1ons,veloci1es

– collisiondetec1on– collisionresolu1on

(numericalsolu1ontoODE)

NewtonianPhysicsasfirst-orderODE

•  mo1onofonepar1cle

•  mo1onofmanypar1cles

Par1cleForces

•  gravity•  viscousdamping•  springs&dampers•  Lennard-Jonespoten1als

F = −kpx − kdv

ProxyForces•  behaviorforces:

flockingbirds,schoolingfish,etc.[“Boids”,CraigReynolds,SIGGRAPH1987]

CurlnoiseforproceduralfluidflowR.Bridson,J.Hourihan,M.Nordenstam,Proc.SIGGRAPH2007.

ODENumericalIntegra1on:Explicit(Forward)Euler

dXdt

= f (X)

ΔX = Δt f (X)

imagefrom:

ExplicitEuler:problems

Solu1onspiralsoutwardsnomaUerhowsmallthe1mestep!(althoughsmaller1mestepsares1llbeUer)

Orcanleadtoinstabili1es:

[PhysicallyBasedAnima1onWitkin&Baraff,SIGGRAPHcoursenotes]

Midpoint&Trapezoidmethods

[slideinpartfromMIT-OCW]

Secondorderaccurate;Midpointmethodalsoknownas“secondorderRunge-KuUa”

Code

EXPLICITEULER

MIDPOINTMETHOD

ImplicitEuler

Xn+1 = Xn + h f (Xn )Xn Xn+1

Xn+1 = Xn + h f (Xn+1)xn+1

backward“implicit”Euler

xn+1 = xn + hvn+1

vn+1 = vn + h(−kxn+1m

)

forward“explicit”Euler

xn+1 = xn + hvn

vn+1 = vn + h(−kxnm)

Idea:Usethederiva1veatthedes1na1onProblem:Wedon’tknowthedes1na1onyet!!

RigidBodyDynamics

•  frompar1clestorigidbodies…

Newton-Eulerequa1onsofmo1onNewton’sequa1onsofmo1on

Preliminaries

•  crossproductviaamatrixmul1ply

!a

Kinema1csofRota1on

•  Intui1vely,withscalars:

•  Moregenerally:

ω r

!ω !r

Fr

!rF

v

!r

!P

m

Newton’sLaw

Euler’sLawtopview

sideview

AngularMomentumofaSetofPar1cles

Iner1aTensor

Newton-EulerEqua1onsofMo1on

Upda1ngtheIner1aTensor

Simula1onLoop

foreach1mestep

setup

solveeqnsofmo1on

integrate

linearposi1onlinearvelocityangularorienta1onangularvelocity

RigidBodyCollisions

•  CollisionDetec1on– broadphase:e.g.,AABBs,boundingspheres– narrowphase:detailedchecks

•  CollisionResponse– collisionimpulses– constraintforces:res1ng,sliding,hinges,….

Par1cle-PlaneCollisionshUps://www.scss.tcd.ie/Michael.Manzke/CS7057/cs7057-1516-09-CollisionResponse-mm.pdf

hUps://www.scss.tcd.ie/Michael.Manzke/CS7057/cs7057-1516-09-CollisionResponse-mm.pdf

hUps://www.scss.tcd.ie/Michael.Manzke/CS7057/cs7057-1516-09-CollisionResponse-mm.pdf

hUps://www.scss.tcd.ie/Michael.Manzke/CS7057/cs7057-1516-09-CollisionResponse-mm.pdf

hUps://www.scss.tcd.ie/Michael.Manzke/CS7057/cs7057-1516-09-CollisionResponse-mm.pdf

hUps://www.scss.tcd.ie/Michael.Manzke/CS7057/cs7057-1516-09-CollisionResponse-mm.pdf