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Problema 1:

PROFESOR: DR. ING. ELMER JAVIERCRDOVA ZAPATA

ALUMNO: LUIS FERNANDO MATEOVALDERRAMA

ROBOTICA

EXAMEN

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Hallar la cine!"ica #irec"a #el r$%$" #e &DOF RPPRRR. Prier$ l$' (ar!e"r$' #e Dena)i"*Har"en%er+ , l-e+$ la a"ri #e "ran'/$raci0n 1$$+2nea.

Parmetros de Denavit-Hartenberg

Ar"ic-laci

0n

3i #i ai 4i

5 35 #5 6 67 6 #7 6 869 6 #9 6 6 3 6 6 *

86; 3; 6 6 86& 3& #& 6 6

Se+

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i*5Ai [cos( i) cos( i) sin( i) sin( i)sin( i) a icos ( i)

sin ( i)0

0

cos ( i) cos( i) sin( i) cos( i) a i sin ( i)sin ( i) cos( i ) d i

0 0 1]

i*5Ai T Ma"ri #e Tran'/$raci0n H$$+2nea.

Para el eercici$= -na )e calc-la#$ l$' (ar!e"r$' #e D*H #e ca#a e'la%0n (r$ce#e$' acalc-lar la' a"rice' A.

6A5 [cos( 1) sin (1) 0 0

sin( 1)0

0

cos( 1) 0 00 1 d 1

0 0 1]

5A7 [1 0 0 0

0

0

0

0 1 01 0 d 2

0 0 1 ]7A9 [

1 0 0 0

0

0

0

1 0 0

0 1 d 3

0 0 1]

9A [cos( 4) 0 sin( 4) 0

sin ( 4)0

0

0 cos ( 4) 01 0 0

0 0 1]

A; [cos( 5) 0 sin ( 5) 0

sin ( 5)0

0

0 cos( 5) 01 0 0

0 0 1]

;A& [cos( 6) sin( 6) 0 0

sin ( 6)0

0

cos( 6) 0 00 1 d 6

0 0 1]

T 6A55A77A99AA;;A&U"ilian#$ Ma"la% (ara -l"i(licar la' a"rice'.

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2. Hallar la cinemtica inversa del robot de !"# P$MA %&. Primero los

'armetros de !enavit(Hartenber) * l+e)o la matri, de trans-ormacin

/omo)0nea.

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Parmetros de Denavit-Hartenberg

Ar"ic-laci0n

3i #i ai 4i

5 35 6 6 *86

7 37 #7 a7 69 39 6 a9 86 3 # 6 *

86; 3; 6 6 86& 3& #& 6 6

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P-e'"$ >-e= T 6A55A77A99AA;;A&'e "en#r! >-e:

6A5*5T 5A77A99AA;;A&

5A7*5 6A5*5T 7A99AA;;A&

7A9*5 5A7*5 6A5*5T 9AA;;A&

9A*5 7A9*5 5A7*5 6A5*5T A;;A&

A;*5 9A*5 7A9*5 5A7*5 6A5*5T ;A&

Hallan#$ la' a"rice':

i*5Ai

[

cos( i) cos( i) sin( i) sin( i)sin( i) a icos ( i)

sin ( i)0

0

cos ( i) cos( i) sin( i) cos( i) a i sin ( i)sin ( i) cos( i ) d i

0 0 1

]i*5Ai T Ma"ri #e Tran'/$raci0n H$$+2nea.6A5 [

cos( 1) 0 sin ( 1) 0

sin( 1)0

0

0 cos ( 1) 01 0 0

0 0 1]

5A7 [cos( 2) sin( 2) 0 a 2 cos( 2)sin( 2)

0

0

cos( 2) 0 a 2 sin( 2)0 1 d 2

0 0 1 ]7A9 [

cos( 3) 0 sin( 3) a3cos ( 3)

sin( 3)0

0

0 cos( 3) a 3sin ( 3)1 0 0

0 0 1]

9A

cos( 4) 0 sin ( 4 ) 0sin( 4 )

0

0

0 cos ( 4) 01 0 d 4

0 0 1]

A;

[

cos( 5) 0 sin ( 5) 0

sin( 5)0

0

0 cos( 5) 01 0 0

0 0 1

]

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;A& [cos( 6) sin( 6) 0 0

sin( 6)0

0

cos( 6) 0 00 1 d 6

0 0 1]

U'an#$ Ma"la% (ara= T 6A&:

U'an#$ Ma"la% (ara 1allar la' a"rice' in)er'a':

6A5*5

cos ( 1) sin ( 1 ) 0 00

sin(1)0

0 1 0cos( 1) 0 0

0 0 1

]5A7*5 [

cos( 2) sin( 2) 0 a 2

sin( 2)0

0

cos( 2) 0 00 1 d 20 0 1

]

7

A9

*5

[cos( 3) sin ( 3) 0 a 3

0

sin( 3)0

0 1 0

cos( 3) 0 00 0 1

]9A*5 [

cos ( 4) sin (4 ) 0 0

0

sin( 4)0

0 1 d 4cos( 4) 0 0

0 0 1]

A;*5 [cos( 5) sin ( 5) 0 0

0

sin( 5)0

0 1 0

cos ( 5) 0 00 0 1]

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;A&*5 [ cos( 6) sin ( 6) 0 0

sin( 6)0

0

cos( 6) 0 00 1 d 60 0 1

]C$n'i#eran#$ Ma"ri in)er'a #e la a"ri 1$$+2nea #e "ran'/$raci0n:

T*5 [nx ny nz nTpox

ax

0

oy oz oTpay az aTp0 0 1

] [nx ny nz p x

ox

ax

0

oy oz py

ay az pz

0 0 1]

En"$nce':

T

[nx ny nz p x

ox

ax0

oy oz py

ay az pz0 0 1

]

*

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U'a$': 6A5*5T 5A77A99AA;;A&

[ cos( 1) sin ( 1 ) 0 0

0

sin(1)0

0 1 0cos( 1) 0 0

0 0 1][

nx ny nz p x

ox

ax

0

oy oz py

ay az pz

0 0 1]=[

cos ( 2 ) sin ( 2 ) 0 a 2cos ( 2 )

sin ( 2 )0

0

cos ( 2 ) 0 a2sin ( 2 )0 1 d 2

0 0 1][

cos ( 3 ) 0 sin ( 3 ) a3cos ( 3 )

sin ( 3 )0

0

0 cos ( 3) a 3 sin( 3 )1 0 0

0 0 1]

[cos( 4) 0 sin ( 4 ) 0sin( 4 )

00

0 cos ( 4) 0

1 0 d 40 0 1

] [cos ( 5 ) 0 sin ( 5 ) 0

sin (5 )

0

0

0 cos ( 5 ) 0

1 0 0

0 0 1

][cos( 6) sin( 6) 0 0sin( 6)

00

cos( 6) 0 0

0 1 d 60 0 1

][

cos( 1) sin ( 1 ) 0 00

sin(1)0

0 1 0cos( 1) 0 0

0 0 1][

nx ny nz p x

ox

ax

0

oy oz py

ay az pz

0 0 1]

T$an#$ ($r ee(l$ eleen"$ 9=:*'in35 P? c$'35 P, #7 #&'in3 'in3;

U'a$': 5

A7*5

6

A5*5

T7

A99

A

A;;

A&

[ cos( 2) sin( 2) 0 a 2

sin( 2)0

0

cos( 2) 0 00 1 d 20 0 1

][ cos( 1) sin ( 1 ) 0 0

0

sin( 1)0

0 1 0cos( 1) 0 0

0 0 1][

nx ny nz p x

ox

ax

0

oy oz py

ay az pz

0 0 1]= [

cos ( 3 ) 0 sin (3 ) a 3cos ( 3 )

sin ( 3 )0

0

0 cos ( 3 ) a 3sin ( 3 )1 0 0

0 0 1]

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[cos( 4) 0 sin ( 4 ) 0sin( 4 )

0

0

0 cos ( 4) 01 0 d 4

0 0 1] [

cos ( 5 ) 0 sin ( 5 ) 0

sin ( 5 )0

0

0 cos ( 5 ) 01 0 0

0 0 1][

cos( 6) sin( 6) 0 0sin( 6)

0

0

cos( 6) 0 00 1 d 6

0 0 1]

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[ cos( 2) cos( 1) cos( 2)sin ( 1) sin(2) a 2sin(2) cos( 1)

s ( 1)0

sin( 2) sin( 1) cos( 2) 0cos( 1) 0 d 2

0 0 1][

nx ny nz p x

ox

ax

0

oy oz py

ay az pz

0 0 1]=

T$an#$ ($r ee(l$ eleen"$ 9=:

'in35 P? c$'35 P, #7 #&'in3'in3;

U'a$': 7A9*5 5A7*5 6A5*5T 9AA;;A&

[cos( 3) sin ( 3) 0 a 3

0

sin( 3)0

0 1 0

cos( 3) 0 00 0 1

] [ cos( 2) sin( 2) 0 a 2

sin ( 2)0

0

cos( 2) 0 00 1 d 20 0 1

][ cos( 1) sin ( 1 ) 0 0

0

sin( 1)0

0 1 0cos( 1) 0 0

0 0 1][

nx ny nz p x

ox

ax

0

oy oz py

ay az pz

0 0 1]

[cos(4) 0 sin ( 4 ) 0sin( 4 )

0

0

0 cos( 4) 01 0 d 4

0 0 1][

cos ( 5 ) 0 sin ( 5 ) 0

sin ( 5 )0

0

0 cos ( 5) 01 0 0

0 0 1][

cos( 6) sin( 6) 0 0sin( 6)

0

0

cos( 6) 0 00 1 d 6

0 0 1]

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cos 3cos 2cos ( 1 )sin 3sin ( 2) cos( 1)

[cos 3cos( 2)sin ( 1 )sin( 3)sin( 2)sin( 1) cos ( 3 ) sin ( 2 )sin( 3)cos( 2) a 2cos ( 3 )a 3 sin ( 1)

(1 )+ sin( 3)cos( 2)cos 0

cos( 3) sin( 2) cos( 1)cos(1)

sin( 3)cos( 2)sin ( 1 )+cos(3)sin( 2) sin( 1) s( 30

[nx ny nz p x

ox

ax

0

oy oz py

ay az pz

0 0 1]=

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T$an#$ ($r ee(l$ eleen"$ 7=:

*'in35 P? c$'35 P, #7 #&'in3'in3;