RigidBodyDynamics

•  frompar2clestorigidbodies…

Newton-Eulerequa2onsofmo2onNewton’sequa2onsofmo2on

SIETE K 294At

stateposition

state Effi EvelueityqorR oriental

I II RangeHEityZ Mai

cnn.IEHeeIcom a EtEIH

IEII.IEtt3etri'iIietgahEtEn EntratforgariotisEI Tx Iaf Terms

Preliminaries

•  crossproductviaamatrixmul2ply

!aa'xtf

abf.bgb'xa axIba axbrotationmatrix

jwJob

biHill Hell IR kfog i.j j k i.toFew in ixj kunit SOG Specialorthogonal

or detCR 1 Iquaternionq w x y E C1124 complexnumberrepresentation

1911 1 unique.ae9emIiegfqtmgandqareathmeOrient

Kinema2csofRota2on

•  Intui2vely,withscalars:

•  Moregenerally:

ω r

!ω !r

Fr

!rF

v

!r

!P

m

IhearMomentum

w r F m fCragen

I r F

Tutor Lab

Fx E Tripangularmomentum

Newton’sLaw

I mtor

a dIIPitot op Fat

Euler’sLawtopview

sideview

inertia temnsen3 3matrixangularvelocity

IF HittanguYpeed

Igf IEEE'Yation

Effluent dat FeiEtta oI E of

AngularMomentumofaSetofPar2cles FE

E TrixieE Exmoi

1

F ri x m Exit

Iri miniEg m riFut

I 3 3 inertia Herson

I ut w I I

Inter2aTensor

F Miri ri wtme.in eitiIEiEitiitimiqzi2ty

2 XiYi Xi2iXYi 3yi.z.x.ie yizi

IXizi Xi'tYi

IForan objectwithDyp symmetries

eg rectangularblade I IyyIzz

Newton-EulerEqua2onsofMo2onNewton IF daff DID

Itt m J

Om I

Euler ET DLIt

d Iafat

IE t IFIT IE t II

Thereegns only hold inan inertial frame

Coriolis term

Non accelerating

Upda%ngtheIner%aTensor

ng

JK 1 Need to compute I in the world frame

40 11 wakeArtis

I gjknownIw RETj RI q RWc

into Ew RI LIRIN Ww compare toIw R Ie R wIw

Simula%onLoop

foreach%mestep

setup

solveeqnsofmo%on

integrate

linearposi%onlinearvelocityangularorienta%onangularvelocity

state FIR rotationmatrix Vq quaternion

I angularmomentum

computeappliedforcesRtorques compileforces torquesinworldframe

Iw RILREF _ha Coriolis EFF

raidedI twxII E

F x trot Fxtvotvvtv.at V t.veR RtikottwhereR wRq qtotwWt.wot Eat

8 0.5 i q w I Lwhhene