Kinematics and geometry review. Based on figures from J Xiao

Transcription

1 Knematcs and geometr ree Based on fgres from J Xao

2 Otlne ee obot Manlators obot onfgraton obot ecfcaton Nmber of Aes DOF recson eeatablt Knematcs relmnar World frame ont frame end-effector frame otaton Matr comoste rotaton matr Homogeneos Matr Drect nematcs Denat-Hartenberg eresentaton Eamles Inerse nematcs

3 Manlators obot arms ndstral robot gd bodes (lns) connected b onts Jonts: reolte or rsmatc Dre: electrc or hdralc End-effector (tool) monted on a flange or late secred to the rst ont of robot

4 obot onfgraton: Manlators artesan: lndrcal: hercal: Artclated: AA: (electe omlance Assembl obot Arm) Hand coordnate: n: normal ector; s: sldng ector; a: aroach ector normal to the tool montng late

5 Manlators Moton ontrol Methods ont to ont a seqence of dscrete onts sot eldng c-and-lace loadng & nloadng ontnos ath follo a rescrbed ath controlled-ath moton ra antng Arc eldng Glng

6 Manlators obot ecfcatons Nmber of Aes Maor aes (-3) > oston the rst Mnor aes (4-6) > Orent the tool edndant (7-n) > reachng arond obstacles aodng ndesrable confgraton Degree of Freedom (DOF) Worsace aload (load caact) recson.s. eeatablt Whch one s more mortant?

7 What s Knematcs Forard nematcs Gen ont arables End-effector oston and orentaton. ) ( n q q q q q q q q L ) ( T A O Y

8 What s Knematcs Inerse nematcs End effector oston and orentaton Jont arables -Formla? ) ( n q q q q q q q q L ) ( T A O

9 Eamle Forard nematcs l l sn l Inerse nematcs ( / l)

10 relmnar obot eference Frames World frame Jont frame Tool frame T W

11 relmnar oordnate Transformaton eference coordnate frame OXYZ Bod-attached frame O ont reresented n OXYZ: T [ ] r + + r + + ont reresented n O : To frames concde > O O

12 relmnar Mtall erendclar Unt ectors roertes of orthonormal coordnate frame roertes: Dot rodct Let and be arbtrar ectors n and be the angle from to then 3

13 relmnar oordnate Transformaton otaton onl r + + r + + Ho to relate the coordnate n these to frames?

14 relmnar Basc otaton and reresent the roectons of onto OX OY OZ aes resectel nce

15 relmnar Basc otaton Matr otaton abot -as th ot ) (

16 relmnar Is t Tre? otaton abot as th sn sn sn sn +

17 Basc otaton Matrces otaton abot -as b otaton abot -as b otaton abot -as b ot ) ( ot() ) ( ot

18 relmnar Basc otaton Matr Obtan the coordnate of from the coordnate of Q T Q 3 I Q T < 3X3 dentt matr Dot rodcts are commtate!

19 Eamle 2 A ont a (432) s attached to a rotatng frame the frame rotates 6 degree abot the OZ as of the reference frame. Fnd the coordnates of the ont relate to the reference frame after the rotaton. a ot( 6) a

20 Eamle 3 A ont a (432) s the coordnate.r.t. the reference coordnate sstem fnd the corresondng ont a.r.t. the rotated OU-V-W coordnate sstem f t has been rotated 6 degree abot OZ as. a T ot( 6) a

21 omoste otaton Matr A seqence of fnte rotatons matr mltlcatons do not commte rles: f rotatng coordnate O-U-V-W s rotatng abot rncal as of OXYZ frame then re-mltl the reos (resltant) rotaton matr th an arorate basc rotaton matr f rotatng coordnate OUVW s rotatng abot ts on rncal aes then ost-mltl the reos (resltant) rotaton matr th an arorate basc rotaton matr

22 Eamle 4 Fnd the rotaton matr for the follong oeratons: ost-mltl f rotate abot the OUVW aes re-mltl f rotate abot the OXYZ aes... abot OU as otaton abot OW as otaton abot OY as otaton Anser + + ot ot I ot - ) ( ) ( ) ( 3

23 oordnate Transformatons oston ector of n {B} s transformed to oston ector of n {A} descrton of {B} as seen from an obserer n {A} otaton of {B} th resect to {A} Translaton of the orgn of {B} th resect to orgn of {A}

24 oordnate Transformatons To ecal ases A A B A o' r r + r B. Translaton onl Aes of {B} and {A} are arallel A B 2. otaton onl Orgns of {B} and {A} are concdent A ' r o

25 Homogeneos eresentaton oordnate transformaton from {B} to {A} Homogeneos transformaton matr o' A B B A A r r r + 3 ' B o A B A A r r r ' r T o A B A B A oston ector otaton matr calng

26 Homogeneos Transformaton ecal cases. Translaton 2. otaton 3 3 B A B A T 3 ' 3 3 o A B A r I T

27 Eamle 5 Translaton along Z-as th h: ) ( h h Trans + h h O O h O O

28 Eamle 6 otaton abot the X-as b ) ( ot

29 Homogeneos Transformaton omoste Homogeneos Transformaton Matr les: Transformaton (rotaton/translaton).r.t (XYZ) (OLD FAME) sng remltlcaton Transformaton (rotaton/translaton).r.t (UVW) (NEW FAME) sng ostmltlcaton

30 Eamle 7 Fnd the homogeneos transformaton matr (T) for the follong oeraton: : abot OZas otaton of d along OZ as Translaton of along OX as a Translaton of abot OX as otaton Anser 4 4 I T T T T T a d a d

31 Homogeneos eresentaton A frame n sace (Geometrc Interretaton) ) ( a s n a s n a s n F n s a F rncal as n.r.t. the reference coordnate sstem

32 Homogeneos Transformaton Translaton n s a n s a ne d a s n d a s n d a s n a s n a s n a s n d d d F old ne F d d d Trans F ) (

33 Homogeneos Transformaton omoste Homogeneos Transformaton Matr A A2 A? Transformaton matr for adacent coordnate frames A 2 A A 2 han rodct of sccesse coordnate transformaton matrces

34 Orentaton eresentaton F otaton matr reresentaton needs 9 elements to comletel descrbe the orentaton of a rotatng rgd bod. An eas a? Eler Angles eresentaton

35 Orentaton eresentaton Eler Angles eresentaton ( ) ψ Man dfferent tes Descrton of Eler angle reresentatons Eler Angle I Eler Angle II oll-tch-ya eqence abot OZ as abot OZ as abot OX as of abot OU as abot OV as abot OY as ψ ψ otatons abot OW as abot OW as abot OZ as ψ

36 Eler Angle I Anmated ' '" " ϕ '" " ' '" ' "

37 Orentaton eresentaton Eler Angle I sn sn sn sn sn sn '' ' ϕ ϕ ϕ ϕ ϕ

38 Eler Angle I + + ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ sn sn sn sn sn sn sn sn sn sn sn sn sn sn ' '' esltant eleran rotaton matr:

39 Eler Angle II Anmated ' "' " ϕ "' ' " "' Note the ooste (clocse) sense of the thrd rotaton. ' "

40 Orentaton eresentaton Matr th Eler Angle II sn snϕ + ϕ snϕ + sn ϕ ϕ sn sn ϕ sn ϕ ϕ sn ϕ snϕ sn sn snϕ sn Q: Ho to get ths matr?

41 Orentaton eresentaton Descrton of oll tch & Ya Z X ϕ Y Q: Ho to get rotaton matr?