PRN 577: Industrial Robotics ASSIGNMENT-3: ASSIGNMENT3: Kinematics Kinem atics
1. Suppose Suppose that that inst instead ead of a frame frame,, a point point P P =(3,5,7,) =(3,5,7,)T in space was translated a distance of d=(2,3,4,)T. ind the new location of the point relati!e to the reference frame. 2. The follo followin win" " was mo!ed mo!ed a distan distance ce of d=(5,2 d=(5,2,#,) ,#,) T$ % 1 % 2 &= 1 % % 4 % % '1 # % % % 1 ind the new location of the frame relati!e to the reference frame. 3. or the followin" followin" frame, frame, find the the !alues of the missi missin" n" elements elements and complete complete the matri matri representation of the frame$ % '1 5 = % % 3 '1 % 2 % % % 1 4. *eri!e the matri matri that repres represents ents a pure rotation rotation a+out a+out the ythe y-ais ais of the reference frame. 5. *eri!e the matri matri that repres represents ents a pure rotation rotation a+out a+out the zthe z-ais ais of the reference frame. #. erif- that that the rotations rotations matrices matrices a+out a+out the reference reference frame frame aes follow follow the reuired reuired constraint euations set +- ortho"onalit- and len"th reuirements of directional unit !ectors. 7. ind ind the the coordi coordinat nates es of point point P P (2, 3, 4) T relati!e to the reference frame after a rotations of 45/ a+out the 'ais. 0. ind ind the the coor coordin dinate atess of point point P P (3, 5, 7) T relati!e to the reference frame after a rotations of 45/ a+out the 'ais. . point p in p in space is defined as & P= 5, P= 5, 3, 4 relati!e to frame & and is attached to the ori"in of the reference frame and is parallel to it . ppl- the followin" transformations to frame B and find P . 6sin" the three'dimensional "rid pro!ided in this chapter, plot the transformations and the result and !erif- it. lso !erif- "raphicall- that -ou would not "et the same result if -ou appl- the transformations relati!e to the current frame$ a. ota otate te %/ %/ a+o a+out ut the the 'a 'ai is. s. +. Then translate 3 units a+out the ythe y-ais, ais, # units a+out the zthe z-ais, ais, and 5 units a+out the 'ais. c. Then Then rotat rotatee %/ %/ a+out a+out the 'ais 'ais.. & 1%. poin pointt p in p in space is defined as P= 2, P= 2, 3, 5 relati!e to frame & which is attached to the ori"in of the reference frame and is parallel to it. ppl- the followin" transformations to frame B frame B and find P . 6sin" the three'dimensional "rid, plot the transformations and the result and !erif- it$ a. ota otate te %/ %/ a+o a+out ut the the 'a 'ai is. s. +. Then, rotate %/ a+out the local a'ais. c. Then Then tran transla slate te 3 units units a+out a+out the the y y-ais, ais, # units a+out the zthe z-ais, ais, and 5 units a+out the 'ais.
11. Show that rotation materials a+out the y'ais and the z 'ais are unitar-. 12. 8alculate the in!erse of the followin" transformations materials$
T1 =
%.527 '%.574 %.#20 2 %.3# %.01 %.43 5 '%.7## % %.#43 3 % % % 1
T2 =
%.2 % %.3 5 % 1 % # '%.3 % %.2 2 % % % 1
13. 9rite the correct seuence of mo!ements that must +e made in order to :unrotate; The spherical coordinates and to ma
Tsph =
1 % % %
% 1 % %
% 3.1375 % 2.15 1 3.214 % 1
(a) ind the necessar- !alues of r, β, γ to achie!e this location. (+) ind the components of the ori"inal matri !ectors for the hand +efore it was unrotated. 15. Suppose that a ro+ot is made of a 8artesian and >? com+inations of @oints. ind the necessar- >? an"les to achie!e the followin"$ %.527 '.574 %.#20 4 T = %.3# %.01 %.43 # '%.7## % %.#43 % % % 1 1#. Suppose that a ro+ot is made of a 8artesian and Auler com+ination of @oints. ind the necessar- Auler an"les to achie!e the followin"$ %.527 T= %.3# '%.7## %
'%.574 %.#20 %.01 %.43 % %.#43 % %
4 # 1
17. frame B was mo!ed alon" its own n'ais a distance of 5 units and then rotated a+out its o' ais an an"les of #%B , followed +- a rotation of a+out 'aisC it was than translated a+out itDs a'ais 3 unites and finall- rotated a+out 'ais 45B. a. 8alculate the total transformations performed. +. 9hat an"les and mo!ements would we ha!e to ma? confi"uration 10. rames descri+in" the +ase of a ro+ot and an o+@ect are "i!en relati!e to the uni!erse frame.
a. ind a transformations TE of the ro+ot confi"uration if the hand of the ro+ot is to +e placed on the o+@ect. +. &- inspection, show whether this ro+ot can +e a three'ais spherical ro+ot, and, if so, find F, G, r. c. ssumin" that the ro+ot is a si'ais ro+ot with 8artesian and Auler coordinates, find , -, , H, I, J.
6
To+@ =
1 % % %
% % 1 %
% '1 % %
1 4 % 1
6
T =
% 1 % %
'1 % % %
% 2 % '1 1 % % 1
1. ro+ot arm with three de"ree s of freedom has +een desi"ned for appl-in" paints on walls as show in fi"ure >.2.1. a. ssi"n coordinate frame as necessar- +ased on E'* representation. +. ill out the parameters the ta+le. c. ind the 6TE matri.
flat
2%. The ro+ot show in fi"ure >.2.2% has two de"rees of freedom, and the transformation matri T is "i!en in s-m+olic form, as well as in numerical form for a specific location. The len"th of each lin< l1and l2 is 1 ft. a. *eri!e the in!erse
21. or the S8't-pe ro+ot show in fi"ure >.2.21, a. ssi"n the coordinate frames +ased on E'* representation. +. ill out the parameters the ta+le. c. 9rite all the matrices. d. 9rite the 6TE matri in terms of the matrices.
22. special three'de"ree'of'freedom spra-in" ro+ot has +een desi"ned as show in fi"ure >.2.22. a. ssi"n the coordinate frames +ased on E'* representation. +. ill out the parameters the ta+le. c. 9rite all the matrices. d. 9rite the T matri in terms of the matrices.
23. or the 6nimation puma 5#2, si'ais ro+ot shown in fi"ure >.2.23, a. ssi"n the coordinate frames +ased on E'* representation. +. ill out the parameters the ta+le. c. 9rite all the matrices. d. ind the 6TE matri for the followin" !alues$ &ase hei"ht =27 in, d =# in , a = 15 in, a =1 in, d =10 in, d =5 in, I1 =% , I2=45 , I3 =% , I4 = % , I5 = ' 45 , I# = % . 24. or the four'de"ree'of'freedom ro+ot depicted in fi"ure >.2.24, a. ssi"n appropriate frames for *'E representation. +. ill out parameters ta+le. c. 9rite an euation in term of matrices that shows how 6TE can +e calculated i". p.2.24
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