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ME 439 Professor N. J. Ferrier
Dynamics of Serial Manipulators
Professor Nicola Ferrier
ME Room 2246, 265-8!"ferrier#en$r%&isc%e'u
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ME 439 Professor N. J. Ferrier
Dynamic Mo'elin$
( 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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ME 439 Professor N. J. Ferrier
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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ME 439 Professor N. J. Ferrier
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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ME 439 Professor N. J. Ferrier
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 $ien y
* &.ere is t.e set of material pointst.at comprise t.e o'y
( 9.e center of mass is
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ME 439 Professor N. J. Ferrier
:nertia
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7/39ME 439 Professor N. J. Ferrier
( Ne&ton-Euler approac.
* P is asolute linear momentum
* F is resultant e0ternal force
* Mo is resultant e0ternal moment &rt point o
* ;o is moment of momentum &rt point o
( a$ran$ian 3ener$y met.o's
Euations of Motion
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8/39ME 439 Professor N. J. Ferrier
( a$ran$ian usin$ $enerali
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9/39ME 439 Professor N. J. Ferrier
Euations of Motion
( a$ran$ian Dynamics, continue'
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10/39ME 439 Professor N. J. Ferrier
Euations of Motions
( Rootics te0ts &ill use eit.er met.o' to'erie euations of motion
* :n =ME "!) >'ance' Rootics an'>utomation? &e use a a$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$.t
for&ar'
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Manipulator Dynamics
( :solate eac. lin7
* Nei$.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
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12/39ME 439 Professor N. J. Ferrier
Notation
9.e follo&in$ are &%r%t% reference frame R)
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13/39ME 439 Professor N. J. Ferrier
Force on :solate' in7
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14/39ME 439 Professor N. J. Ferrier
9orue on :solate' in7
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15/39ME 439 Professor N. J. Ferrier
e0ternal
>pplie' y
actuators in< 'irection
Force-torue alance on manipulator
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16/39ME 439 Professor N. J. Ferrier
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
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17/39ME 439 Professor N. J. Ferrier
Force+9orue on lin7 n
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18/39ME 439 Professor N. J. Ferrier
Ne&ton@s a&
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19/39ME 439 Professor N. J. Ferrier
Ne&ton-Euler >l$orit.m
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20/39ME 439 Professor N. J. Ferrier
Ne&ton-Euler >l$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
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21/39ME 439 Professor N. J. Ferrier
=.an$e of coor'inates? forforce+torue
R i N t E l
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22/39ME 439 Professor N. J. Ferrier
Recursie Ne&ton-Euler>l$orit.m
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23/39ME 439 Professor N. J. Ferrier
9&o-lin7 manipulator
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24/39ME 439 Professor N. J. Ferrier
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
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For&ar' inematics) planar 2-lin7 arm
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26/39ME 439 Professor N. J. Ferrier
For&ar' inematics) planar 2-lin7 manipulator
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For&ar' inematics) planar 2-lin7 manipulator
&%r%t% aseframe GH
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ME 439 Professor N. J. Ferrier
For&ar' inematics) planar 2-lin7 manipulator
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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ME 439 Professor N. J. Ferrier
For&ar' inematics) planar 2-lin7 manipulator
&%r%t% ase frame GH
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ME 439 Professor N. J. Ferrier
Point Mass mo'el for t&o lin7 planar arm
D; tale for t&o lin7 arm
mA m2
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ME 439 Professor N. J. Ferrier
Dynamic Mo'el of 9&o in7 >rm&+point mass
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ME 439 Professor N. J. Ferrier
1eneral Form
Iointtorues
:nertia3mass
Iointaccelerations
1raityterms
oriolis Jcentripetalterms
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ME 439 Professor N. J. Ferrier
1eneral Form) No motion
Ioint toruesreuire' to .ol'manipulator in a
static position 3i%e%
counter$raitational forces
1raity terms
No motion so
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ME 439 Professor N. J. Ferrier
:n'epen'ent Ioint ontrol reisite'
( alle' =ompute' 9orue Fee'for&ar'?in te0t
( Kse 'ynamic mo'el / setpoints 3'esire'position, elocity an' acceleration from
7inematics+traectory plannin$ as afee'for&ar' term
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ME 439 Professor N. J. Ferrier
Manipulator motion from input torues
:nte$rate to $et
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ME 439 Professor N. J. Ferrier
Dynamic Mo'el of 9&o in7 >rm&+point mass
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ME 439 Professor N. J. Ferrier
Dynamics of 2-lin7 * point mass
Dynamics in loc7 'ia$ram of 2 lin7 3point mass
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ME 439 Professor N. J. Ferrier
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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