at rigid body dynamics - university of british columbia
TRANSCRIPT
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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
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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
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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
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Newton’sLaw
I mtor
a dIIPitot op Fat
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Euler’sLawtopview
sideview
inertia temnsen3 3matrixangularvelocity
IF HittanguYpeed
Igf IEEE'Yation
Effluent dat FeiEtta oI E of
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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
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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
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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
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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
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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