Ferrier SMdynamics

8/17/2019 Ferrier SMdynamics

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ME 439 Professor N. J. Ferrier

Dynamics of Serial Manipulators

Professor Nicola Ferrier

ME Room 2246, 265-8!"ferrier#en$r%&isc%e'u


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Dynamic Mo'elin$

( For manipulator arms)

* Relate forces+torues at oints to t.e motionof manipulator / loa'

( E0ternal forces usually only consi'ere' at t.een'-effector

( 1raity 3lift arms is a maor consi'eration


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Dynamic Mo'elin$

( Nee' to 'erie t.e euations of motion

* Relate forces+torue to motion

( Must consi'er 'istriution of mass

( Nee' to mo'el e0ternal forces


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Manipulator in7 Mass

( onsi'er lin7 as a system of particles

* Eac. particle .as mass, 'm

* Position of eac. particle can e e0presse'usin$ for&ar' 7inematics


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Manipulator in7 Mass( 9.e 'ensity at a position 0 is ρ30,

* usually ρ is assume' constant

( 9.e mass of a o'y is $ien y

* &.ere is t.e set of material pointst.at comprise t.e o'y

( 9.e center of mass is


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:nertia


7/39ME 439 Professor N. J. Ferrier

( Ne&ton-Euler approac.

* P is asolute linear momentum

* F is resultant e0ternal force

* Mo is resultant e0ternal moment &rt point o

* ;o is moment of momentum &rt point o

( a$ran$ian 3ener$y met.o's

Euations of Motion



( a$ran$ian usin$ $enerali



Euations of Motion

( a$ran$ian Dynamics, continue'



Euations of Motions

( Rootics te0ts &ill use eit.er met.o' to'erie euations of motion

* :n =ME "!) >'ance' Rootics an'>utomation? &e use a a$ran$ian approac.usin$ computational tools from 7inematics

to 'erie t.e euations of motion

( For simple roots 3planar t&o lin7 arm,Ne&ton-Euler approac. is strai$.t

for&ar'



Manipulator Dynamics

( :solate eac. lin7

* Nei$.orin$ lin7s apply e0ternal forces an'torues

( Mass of nei$.orin$ lin7s

( E0ternal force in.erite' from contact et&een tip

an' an oect( D@>lemert force 3if nei$.orin$ lin7 is

acceleratin$

* >ctuator applies eit.er pure torue or pure

force 3y D; conention alon$ t.e



Notation

9.e follo&in$ are &%r%t% reference frame R)



Force on :solate' in7



9orue on :solate' in7



e0ternal

>pplie' y

actuators in< 'irection

Force-torue alance on manipulator



Ne&ton@s a&

( > net force actin$ on o'y pro'uces a

rate of c.an$e of momentum inaccor'ance &it. Ne&ton@s a&

( 9.e time rate of c.an$e of t.e totalan$ular momentum of a o'y aout t.eori$in of an inertial reference frame iseual to t.e torue actin$ on t.e o'y



Force+9orue on lin7 n



Ne&ton@s a&



Ne&ton-Euler >l$orit.m



Ne&ton-Euler >l$orit.m

A% ompute t.e inertia tensors,

2% Bor7in$ from t.e ase to t.e en'-effector, calculate t.e positions,elocities, an' accelerations of t.e

centroids of t.e manipulator lin7s &it.respect to t.e lin7 coor'inates37inematics

"% Bor7in$ from t.e en'-effector to t.e

ase of t.e root, recursiely calculatet.e forces an' torues at t.e actuators&it. respect to lin7 coor'inates

=. f 'i ? f



=.an$e of coor'inates? forforce+torue

R i N t E l



Recursie Ne&ton-Euler>l$orit.m



9&o-lin7 manipulator



9&o lin7 planar arm

C θA

D; tale for t&o lin7 arm

0 02

C2

in7 ar θ ' α a

A θA θA A

2 θ2 θ2 2

A 2

0A

CA

θ2



For&ar' inematics) planar 2-lin7 arm



For&ar' inematics) planar 2-lin7 manipulator




&%r%t% aseframe GH


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position ector from ori$in offrame to c%o%m% of lin7 Ae0presse' in frame

position ector from ori$in

of frame A to c%o%m% of lin7 2e0presse' in frame

position ector from ori$inof frame A to ori$in of frame2 e0presse' in frame

position ector from ori$in offrame to ori$in of frame A

e0presse' in frame


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&%r%t% ase frame GH


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Point Mass mo'el for t&o lin7 planar arm

D; tale for t&o lin7 arm

mA m2


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Dynamic Mo'el of 9&o in7 >rm&+point mass


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1eneral Form

Iointtorues

:nertia3mass

Iointaccelerations

1raityterms

oriolis Jcentripetalterms


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1eneral Form) No motion

Ioint toruesreuire' to .ol'manipulator in a

static position 3i%e%

counter$raitational forces

1raity terms

No motion so


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:n'epen'ent Ioint ontrol reisite'

( alle' =ompute' 9orue Fee'for&ar'?in te0t

( Kse 'ynamic mo'el / setpoints 3'esire'position, elocity an' acceleration from

7inematics+traectory plannin$ as afee'for&ar' term


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Manipulator motion from input torues

:nte$rate to $et


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Dynamic Mo'el of 9&o in7 >rm&+point mass


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Dynamics of 2-lin7 * point mass

Dynamics in loc7 'ia$ram of 2 lin7 3point mass


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Dynamics in loc7 'ia$ram of 2-lin7 3point mass

Dynamics of 2 lin7 slen'er ro'


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Dynamics of 2-lin7 * slen'er ro'

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Ferrier SMdynamics