135750458-Robot-Dynamics-Newton-Euler-Recursive-ApproachP3.ppt

8/16/2019 135750458-Robot-Dynamics-Newton-Euler-Recursive-ApproachP3.ppt

1/27

Robot Dynamics – Newton- EulerRecursive Approach

ME 4135 Robotics &

Controls

R. Lindeke, Ph. D.


2/27

Physical Basis:

This method is jointl b!sed on" # $e%tons 'nd L!% o( Motion

E)*!tion"

!nd considerin+ ! ri+id link

# E*lers -n+*l!r orce/ Moment

E)*!tion"

i C F mν = &

i imoment CM i i CM i N I I ω ω ω = + ×&


3/27

Again we will Fin A !"or#ue$ %oel

E!ch Link 0ndiid*!ll

2e %ill moe (rom !se to End to (ind elocities!nd -cceler!tions

2e %ill moe (rom End to !se to com*te (orce

6(7 !nd Moments 6n7

in!ll %e %ill (ind th!t the Tor)*e is"

( ) ( ) ( )1 1(1 )i i iqT T

n z f z bi i i i i i

τ ξ ξ − −= + − + &

8r!it is imlicitl incl*ded in the model b considerin+

!cc9 : + %here + is 69, ;+9, 97 or 69, 9, ;+97

i is the joint te!r!meter 6is 1 i(

reol*te< 9 i(

rism!tic7 like in

=!cobi!n>


4/27

&ets &oo' at a &in' !%oel$


5/27

(e will Buil )elocity E#uations

Consider th!t i is the joint

te !r!meter 6is 1 i(

reol*te< 9 i( rism!tic7

-n+*l!r elocit o( ! r!me

k relative to the Base"

$?TE" i( joint k is rism!tic,the !n+*l!r elocit o( (r!me

k is the s!me !s !n+*l!r

elocit o( (r!me k;1>

1 1k k k k k q z ω ω ξ − −= + &


6/27

Angular Acceleration o* a !Frame$

T!kin+ the Time Deri!tie o( the !n+*l!r elocit

model o( r!me k"

( )1 1 1 1k k k k k k k k q z q z ω ω ξ ω − − − − = + + × & & && &

+ame as ,w/t the

angular acceleration in

ynamics


7/27

&inear )elocity o* Frame ':

De(inin+ sk : dk # dk;1 !s ! link ector, Then the

linear velocity o( link @ is"

Le!din+ to ! Linear Acceleration Model o("

( ) ( )1 11k k k k k k k v v s q z ω ξ − −= + × ∆ + − &

( )

( ) ( )

1

1 11 2

k k k k k k k

k k k k k k

s s

q z q z

ν ν ω ω ω

ξ ω

−

− −

= + × ∆ + × × ∆

+ − + ×

& & &

&& &

$orm!l comonent o(

!cceler!tion 6centri(*+!l

!cceler!tion7


8/27

"his completes the Forwar Newton-

Euler E#uations:

To e!l*!te Link elocities & !cceler!tions, st!rt %ith the -AE 6r!me97

0ts Aet & - set 6(or ! (iBed or inerti!l b!se7 is"

-s !dertised, settin+ b!se line!r !cceler!tion ro!+!tes +r!it!tion!le((ects thro*+ho*t the !rm !s %e rec*rsiel moe to%!rd the end>

0

0

0

0

00

0v

g

ω ω

ν

==

=

= −

&

&


9/27

Now we e*ine the Bac'war

,Force.%oment E#uations

2ork Rec*rsiel (rom the End

2e de(ine ! term∆

r k %hich is the ector (rom

the end of a link to its center of mass"

k k k

k

r c d

c

∆ = −

is location of center of mass of Link k


10/27

De*ining * an n %oels

( )1k k k k k k k k k f f m r r ν ω ω ω + = + + × ∆ + × × ∆

& &

( )

( )

1 1k k k k k k k k k

k k k

n n s r f r f D

D

ω

ω ω

+ += + ∆ + ∆ × − ∆ × +

+ ×

&

0nerti!l Tensor o( Link k #

in b!se s!ce

The term in the br!ckets reresents

the line!r !cceler!tion o( the center

o( m!ss o( Link k


11/27

/ombine them into "or#ue %oels:

( ) ( ) ( ) ( )1 11T T

k k k k k k k k k n z f z b qτ ξ ξ − −= + − + &

$?TE" or ! robot moin+(reel in its %orks!ce %itho*t

c!rrin+ ! !lo!d, ( tool : 9

2e %ill be+in o*r rec*rsion b settin+ ( n1 : ;( tool !nd nn1: ;ntoolorce !nd moment on the tool


12/27

"he N-E Algorithm:

Ate 1" set T99 : I< ( nC1 : ;( tool< nnC1 : ;ntool< 9 : 9<

dot9 : ;+< 9 : 9< dot9 : 9

Ate '" Com*te # k;1s

-n+*l!r elocit & -n+*l!r -cceler!tion o( Link k

Com*te sk

Com*te Line!r elocit !nd Line!r !cceler!tion o( Link k

Ate 3" set k : k1, i( k:n b!ck to ste ' else set k : n

!nd contin*e


13/27

"he N-E Algorithm cont0:

Ate 4" Com*te #

r k 6rel!ted to center o( m!ss o( Link k7

( k 6(orce on link k7

$k 6moment on link k7tk

Ate 5" Aet k : k;1. 0( kF:1 +o to ste 4

( )0 0T

k k

k k D R D R=


14/27

+o1 &ets "ry one:

@eein+ it

Extremely Simple

This 1;!Bis robot

is c!lled !n

0nerted Pend*l*m

0t rot!tes !bo*t G9

Hin the l!neI


15/27

(riting some in*o about the evice:1

1

2

1 11

1 1 1 1

1 1 1 11

0 1

2

0

01

0 0 0

0 1 012

0 0 1

0

0

0 0 1 0

0 0 0 1

a

c

m a D

C S a C

S C a S T A

−

∆ =

= ÷

−

= =

HLinkI is ! thin

clindric!l rod


16/27

/ontinuing an computing:

1 1 1

1 0

1

1 1 1 1

1 1 1 1

1 1

1 1

0 21 0 0 00 00 1 0 0

0 0 1 0 00 0 1 00 0 0 1 1

2

2

0

c H T c

a

C S a C S C a S

a C

a S

= ∆

− − = • •

=


17/27

2nertial "ensor computation:

( )1 11 0 1 0

1 1 1 12

1 11 1 1 1

2

1 1 1221 1

1 1 1

0 0 0 0 0

0 0 1 0 012

0 0 1 0 0 1 0 0 1

0

012

0 0 1

T

D R D R

C S C S m a

S C S C

S S C m a

S C C

=

− = • • − ÷ −

= − ÷


18/27

&et Do it ,Angular )elocity 3 Accel04

1 1 1

1 1 1

1 1

1 111 0

1 1

1 1

0 0

0 0 0

1 1

0 00 0 0

1 1

1 0 0 0

( ) 0 1 0 00

0 0 1 01

0

o

q q

q q

a C

a S s H O O

a C

a S

ω

ω

= + =

= + =

÷

÷ ∆ = − = • ÷ ÷ ÷

=

& &

& && &&

At!rtin+" !se 6i:97 -n+. el : -n+. !cc :

Lin. el : 9

Lin. -cc : ;+ 69, ;+9, 97T

1 : 1


19/27

&inear )elocity:

( )

( )

1 0 1 1 1 1 0

1

1 1 1 1

1

1 1 1

1

0 00 0 1 1 0

1 0 1

0

v v s q z

C q a S q

S

a q C

ω ξ = + × ∆ + −

÷ = + × + − ÷ ÷

− =

&

& &

&


20/27

&inear Acceleration:

( )1 1 1 1 1 1

1 1

1 1 1 1 1 1 1

1 1

2

1 1 1 1 1 1

1 1

2

1 1 1 1 1 1

0

0

0 1 0

0 0

0 0

g s s

S S

g a q C q a q C

S C

g a q C a q S

S C

g a q C a q S

ν ω ω ω = − + × ∆ + × × ∆

− − = − + + × − −

= − + + − −

= − + −

& &

&& & &

&& &

&& &

Note:

g 5 ,61 -g61 6"


21/27

"hus Forwar Activities are one4

Com*te r 1 to

be+in !ck%!rd

orm!tions"

1 1 1

1 1

1 1

1 11 1

1

11

2

200

2

0

r c O

a C

a C a S a S

C a

S

∆ = −

= −

= −


22/27

Fining * 7

Consider"

( tool : 9

( )1 1 1 1 1 1 1

1 1

1 1 1 1

1 1 1 1 1

1

1 1

1 1 1

{ }

0 0 0

{ 0 0 0 }2 2

1 0 1 1 0

{2

0

f m r r

C C a q a q

m S q S

S a q

m C

ν ω ω ω

ν

ν

= + × ∆ + × × ∆

÷ ÷ ÷ = − × + × − × ÷ ÷ ÷ ÷ ÷ ÷ ÷

− = − − ÷

& &

&& && &

&&&

12

1 1

1

1 12

1 1 1 11 1 1 1

1 1 1 2

2 1 1 1 1

1 1 1 1 1 1 1 1

0

0 }2

1 0

{ }2 2

0 0

2 2

0 0 0

S a q

C

S C

a q a qm C S

S C S a q a q

m g a q C a q S C

ν

− × ÷

−

= − + ÷ ÷ − −

= − + − − + ÷

&

&& &&

& &&& &

1

1

0

C

S

÷

÷ ÷

÷


23/27

/ollapsing the terms

( ) ( )

1 12

1 1 1 11 1 1 1

2 21 1 1 1 1 1 1 1 1 1

1 0

2 20 0

02 2

T

S C a q a q f m g C S

a q S q C a q C q S m g

− ÷ = − + − ÷ ÷

+ −= − +

&& &

&& & && &

Note * 7 is a )ector4


24/27

/omputing n7:

( ) ( )

( ) ( )

1 1 1 1 1 1 1 1 1

1 1 2 2 2

1 1 1 1 1 1 11 1 1 1

2 21

1 1 1 1 1 1 1 1 1 111 1 0

0 0 0

0 0 02 12 12

0 0 1 1 1

02 2 2

0

n s r f D D

C C

a m a q m a qa S S f

C a q S q C a q C q S a

S m g

ω ω ω = ∆ + ∆ × + + ×

÷ = − × + + × ÷ ÷

+ − = × − +

&

&& &

&& & && &

( ) ( )

2

1 1 1

2 2 21 1 1 1 1 1 1 1 1 1 1 1 11 1

1 0 1

0

012

1

0 0

0 02 2 2 12

1 1

T

m a q

a q C q S a q S q C m a qa mC g S

÷ + ÷

− + ÷ ÷ ÷= + − − + ÷ ÷ ÷

&&

&& & && & &&

"his 8-prouct goes to 9ero4

The Link orce ector


25/27

+impli*ying:

21 0 1 1 11 1 1

00

3 2 1

m g m a C n a q

= + ÷

&&


26/27

(riting our "or#ue %oel

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

1 1 1 0 1 1 0 1 1

21 0 1 1 11 1 1 1

2

1 1 0 1 1 11 1 1 1

10 0

0 0 0 1 1 03 2

1 1

3 2

T T

T

n z f z b q

m g m a C a q f b q

m a g m a C q b q

τ ξ ξ

τ

= + − +

= + + − + ÷

= + + ÷

&g g

&& &g g

&& &

Dot 6sc!l!r7 Prod*cts


27/27

omewor' Assignment:

J Com*te L;E sol*tion (or H0nerted Pend*l*m &

Com!re tor)*e model to $;E sol*tion

J Com*te $;E sol*tion (or ' link !rtic*l!tor 6o( slide

set" Dn!mics, !rt '7 !nd com!re to o*r L;E

tor)*e model sol*tion com*ted there

J Consider ?*r 4 !Bis 8M clindric!l robot # i( thelinks c!n be simli(ied to thin clinders, deelo !

+ener!liGed tor)*e model (or the deice.

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