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8/20/2019 Chapter 4 - Dynamic Analysiss
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Chapter 4Dynamic Analysis and Forces
4.1 INTRODUCTION
In this chapters…….
• The dynamics, related with accelerations, loads, masses andinertias.
__ __
am F
⋅=∑
__ __
α
⋅=∑ I T
In Actators……. • The actator can !e accelerate a ro!ot"s lin#s $or e%ertin& eno&h$orces
and tor'es at a desired acceleration and (elocity.
• )y the dynamic relationships that &o(ern the motions o$ thero!ot,
*i&. 4.1 *orce+mass+acceleration and tor'e+inertia+an&lar
acceleration relationships $or a ri&id !ody.
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Chapter 4Dynamic Analysis and Forces
4. -ARANIAN /0CANIC23 A 2ORT O0RI05
• -a&ran&ian mechanics is !ased on the di6erentiationener&y terms only, with respect to the system"s (aria!les and time.• De7nition3 L 8 -a&ran&ian, K 8 9inetic 0ner&y o$ the system, P 8:otential 0ner&y, F 8 the smmation o$ all e%ternal $orces $or alinear
motion, T 8 the smmation o$ all tor'es in a rotationalmotion, x 8 2ystem (aria!les
P K L −=
ii
i
x L
x
Lt
F ∂∂−
∂∂∂∂= ⋅
ii
i
L L
t T
θ θ ∂∂
−
∂
∂∂∂
= ⋅
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Chapter 4Dynamic Analysis and Forces
Example 4.1
*i&. 4. 2chematic o$ a simple cart+sprin&system.
*i&. 4.; *ree+!ody dia&ram $or the sprint+cartsystem.
• -a&ran&ianmechanics
22
2
2
1,
2
1
2
1kx P xmmv K ===
•2
2
2
1
2
1kx xm P K L −=−=
•
• Newtonianmechanics
. . ..
. , ( ) ,
i
L d Lm x m x m x kx
dt x x
∂ ∂= = = −
∂∂
kx xm F += ..
__ __
am F ⋅=∑kxma F makx F +=→=−
• The comple%ity o$ the terms increases as the nm!er o$ de&rees o$$reedom
Solution
Deri(e the $orce+acceleration relationship $or the one+de&ree o$ $reedom system.
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Chapter 4Dynamic Analysis and Forces
Example 4.2
*i&. 4.4 2chematic o$ a cart+pendlm
system.
Solution
Deri(e the e'ations o$ motion $or the two+de&ree o$ $reedom system.
In this system…….
It requires two coordinates, x and .
It requires two equations of motion:
1. The linear motion of the system.
2. The rotation of the pendulum.
+
+
+=
θ θ
θ
θ θ
θ
sin00
sin0
cos
cos
2.2
.22
..
..
222
221
gl m
kx xl m x
l ml m
l mmm
T
F
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Chapter 4Dynamic Analysis and Forces
Example 4.4
*i&. 4.< A two+de&ree+o$+$reedom ro!otarm.
Solution
Usin& the -a&ran&ian method, deri(e the e'ations o$ motion $or thetwo+de&ree o$ $reedom ro!ot arm.
Follow the same steps as before…….
alculates the !elocity of the center of
mass of lin" 2 by differentiatin# its position:
The "inetic ener#y of the total system is the
sum of the "inetic ener#ies of lin"s 1 and 2.
The potential ener#y of the system is the
sum of the potential ener#ies of the two
lin"s:
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Chapter 4Dynamic Analysis and Forces
4.; 0**0CTI0 /O/0NT2 O* IN0RTIA
• To 2impli$y the e'ation o$ motion, 0'ations can !erewritten in sym!olic $orm.
+
=
+
=
j
i
jjj jii
ijjiii
jjj jii
ijjiii
j
i
jj ji
ijii
D
D
D D
D D
D D
D D
D D
D D
T
T .
1
.
2.
2
.
1.22
.21
..
..
2
1
θ
θ
θ
θ
θ
θ
θ
θ
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Chapter 4Dynamic Analysis and Forces
4.4 D=NA/IC 0>UATION2 *OR /U-TI:-0+D0R00+O*+*R00DO/ RO)O
• 0'ations $or a mltiple+de&ree+o$+$reedom ro!ot are (ery lon& andcomplicated, !t can !e $ond !y calclatin& the #inetic and
potentialener&ies o$ the lin#s and the ?oints, !y de7nin& the -a&ran&ian and!y
di6erentiatin& the -a&ran&ian e'ation with respect to the ?oint(aria!les.
4.4.1 9inetic 0ner&y
The "inetic ener#y of a ri#id bodywith motion in three dimension :
GhV m K __
2
2
1
2
1ω +=
The "inetic ener#y of a ri#id bodyin planar motion
22
2
1
2
1ω I V m K +=
*i&. 4.@ A ri&id !ody in three+dimensional motionand
in plane motion.
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Chapter 4Dynamic Analysis and Forces
4.4 D=NA/IC 0>UATION2 *OR /U-TI:-0+D0R00+O*+*R00DO/ RO)O
4.4.1 9inetic 0ner&y
The !elocity of a point alon# a robot$s lin" can be defined by differentiatin#the position equation of the point.
iiii Ri r T r T p 0== The !elocity of a point alon# a robot$s lin" can be defined by differentiatin#
the position equation of the point.
( ) ∑∑∑∑ == = = +=n
i
iact ir
n
i
i
p
i
r
pT ir iipi q I qqU J U Trace K 1
2)(
1 1 1 21
21
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Chapter 4Dynamic Analysis and Forces
4.4 D=NA/IC 0>UATION2 *OR /U-TI:-0+D0R00+O*+*R00DO/ RO)O
4.4. :otential 0ner&y
The potential ener#y of the system is the sum of the potential ener#ies of each lin".
])([1
0
1 ∑∑ == ⋅−==n
i
iiT
i
n
i
i r T g m p P
The potential ener#y must be a scalar quantity and the !alues in the #ra!ity
matrix are dependent on the orientation of the reference frame.
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Chapter 4Dynamic Analysis and Forces
4.4 D=NA/IC 0>UATION2 *OR /U-TI:-0+D0R00+O*+*R00DO/ RO)O
4.4.; The -a&ran&ian
( ) r n
i
i
p
i
r p
T
ir iip qqU J U Trace P K L ∑∑∑= = ==−= 1 1 121
])([2
1
1
0
1
2)( ∑∑
==⋅−−+
n
i
iiT
i
n
i
iact i r T g mq I
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Chapter 4Dynamic Analysis and Forces
4.4 D=NA/IC 0>UATION2 *OR /U-TI:-0+D0R00+O*+*R00DO/ RO)O
4.4.4 Ro!ot"s 0'ations o$ /otion
The %a#ran#ian is differentiated to form the dynamic equations of motion.
The final equations of motion for a #eneral multi&axis robot is below.
i
n
j
n
k
k jijk iact i
n
j
jiji Dqq Dq I q DT ∑∑∑= ==
+++=1 1
)(
1
)(),max(
T pi p
n
ji p
pjij U J U Trace D
∑==)(
),,max(
T pi p
n
k ji p
pjk ijk U J U Trace D ∑=
=
∑=
−=n
i p
p piT
pi r U g m D
where,
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Chapter 4Dynamic Analysis and Forces
Example 4.7
*i&. 4. The two+de&ree+o$+$reedom ro!ot arm o$0%ample 4.4
Solution
Usin& the a$orementioned e'ations, deri(e the e'ations o$ motion $orthe two+de&ree o$ $reedom ro!ot arm. The two lin#s are assmed to !eo$ e'al len&th.
Follow the same steps as before…….
'rite the ( matrices for the two lin"s)
*e!elop the , and for the robot.ij D ijk D i D
The final equations of motion without the actuator inertia terms are the same as below.
22
2
2
2
212
2
2
2
2
2
112
1
3
1
3
4
3
1
θ θ
++
++= C l ml mC l ml ml mT
( ) 1)(1121221121222222222
1
2
1
2
1θ θ θ θ act I glC m glC m glC m l m l m +++++
+
1)(21222
2
22
2
212
2
2
2
22 2
1
2
1
3
1
2
1
3
1θ θ θ
act I glC m l ml mC l ml mT ++
+
+
+=
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Chapter 4Dynamic Analysis and Forces
4.B 2TATIC *ORC0 ANA-=2I2 O*RO)OT2
+osition ontrol: The robot follows a prescribed path without any reacti!e force.
obot ontrol means +osition ontrol and Force ontrol.
Force ontrol: The robot encounters with un"nown surfaces and mana#es tohandle the tas" by ad-ustin# the uniform depth while #ettin# the reacti!e force.
x/ Tappin# a 0ole & mo!e the -oints and rotate them at particular rates tocreate the desired forces and moments at the hand frame.
x/ +e# Insertion a!oid the -ammin# while #uidin# the pe# into the hole andinsertin# it to the desired depth.
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Chapter 4Dynamic Analysis and Forces
4.B 2TATIC *ORC0 ANA-=2I2 O*RO)OT2
To elate the -oint forces and torques to forces and moments #enerated at the hand frame of the robot.
T ! ! ! ! ! ! !
x " # x " # F $ $ $ m m m =
x " # x " # x #
dx
d"
d# % $ $ $ m m m $ dx m # x
"
#
δ δ
= = + + ∂ ∂ ∂
- -
[ ] [ ] [ ] F J T ! T ! =
[ ] [ ] [ ] [ ]θ δ DT D F % T ! T ! ==
f is the force and m is the moment
alon# the axes of the hand frame.
The total !irtual wor" at the -ointsmust be the same as the total wor"at the hand frame.
eferrin# to (ppendix (
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Chapter 4Dynamic Analysis and Forces
4.< TRAN2*OR/ATION O* *ORC02 AND /O/0NT2 )0T500N COORDINAT0*RA/02
(n equi!alent force and moment with respect to the other coordinate frameby the principle of !irtual wor".
[ ] [ ] # " x # " xT
mmm $ $ $ F =[ ] [ ] # " x # " x
T d d d D δ δ δ =
[ ] [ ] # &
" &
x &
# &
" &
x &T &
mmm $ $ $ F =[ ] [ ] # & " & x & # & " & x &
T & d d d D δ δ δ =
[ ] [ ] [ ] [ ] DT D F % &T &T ==δ
The total !irtual wor" performed on the ob-ect in either frame must be the same.
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Chapter 4Dynamic Analysis and Forces
4.< TRAN2*OR/ATION O* *ORC02 AND /O/0NT2 )0T500N COORDINAT0*RA/02
*isplacements relati!e to the two frames are related to each other by thefollowin# relationship.
[ ] [ ][ ] D J D & & =
The forces and moments with respect to frame B is can be calculated directlyfrom the followin# equations:
$ n $ x & ⋅=
$ ' $ " & ⋅=
$ ' $ " & ⋅=
( ) ][ m p $ nm x & +×⋅=
( ) ][ m p $ 'm " & +×⋅=
( ) ][ m p $ am # & +×⋅=
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