Kinematics. Redundancy. Task Redundancy. Operational Coordinates. Generalized Coordinates. m task. Manipulator. Operational point

Transcription

1 Mapulato smatc Jot Revolute Jot Kematcs Base Lks: movg lk fed lk Ed-Effecto Jots: Revolute ( DOF) smatc ( DOF) Geealzed Coodates Opeatoal Coodates O : Opeatoal pot 5 costats 6 paametes { postos oetatos {} movg lks: 6 paametes d.o.f. jots: 5 costats d.o.f. (system): 6-5 = A set,,, m of m depedet cofguato paametes m : umbe of degees of feedom of the ed-effecto. Redudacy ask Redudacy A obot s sad to be edudat f m Degees of edudacy: m m task m task : degees of edudacy/task

2 Homogeeous asfomato Geometc Model Compact epesetato R R R ; 4 4 : ot othoomal Cosecutve asfomatos R R ' k k Kematc Cha jot Fowad Kematcs Ivese Kematcs jot Lk z, z Lk Lk z R Lk y Lk R z jot y Lk R z jot y R y R Lk y

3 Deavt-Hatebeg (DH) aametes jot Homogeeous asfomato, a,, ( ) ( ) z R Lk y R z jot Lk ( ) cos s a( ) s cos cos cos s s s s cos s cos cos ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a a R ( ) Geometc Model Fowad Kematcs ( ) ( ) ( ) ( ) ( ) ( ) ( ) R R R R R Ro ( ) ( ) o ( ) ( ) ( ) ( ) m euatos G ( ) p ( ) ( ) Repesetatos R Catesa Sphecal Cyldcal. Eule Agles Decto Coses Eule aametes

4 osto Repesetatos ( p ) s obtaed fom ( ) ( ) Oetato Repesetato ( R ) s obtaed fom R ( ) ( ) Catesa yz,, Cyldcal,,z Sphecal,, z y R R ( ) R R ( ) Decto Coses R ( ) ( ) ( ) ; ( ) ( ) ( ) ( ) Eule Agles z z z ( ) ( ) R R R R ( ) z( ) z y( ) ( ) y R c cscs csscc ss R ( ) ( ) s cccs ssccc cs s s s c c ( ) ( ) ( ) ( ) ( ) sg accos ; ( ) accos ; Sgulaty of the epesetato ( ) sg accos ; k k:(tege) o ae defed- 4

5 Eule aametes Quateo Rotatos: oduct of two plae symmetes uv cos uv ws w v u Eule 4-aametes cos ; w s ; w s ; w s ; Nomalty codto R Rotato Mat ( ) ( ) R ( ) Sg Detemato:,,,, ad ; sg ; sg ; sg ; Lemma: Fo all otatos, at least oe of the Eule aametes has a magtude lage tha o eual to /. Algothm wth whee 5

6 Eule Agles & aametes cos cos ( ) ; s cos ( ) ; s s ( ) ; cos s ( ). Basc Jacoba {} v FI v GJ HK (6 ) lea velocty agula velocty J ( ) ( 6 ) ( ) Jacoba fo X Gve a epesetato J ( ) R Jacoba ad Basc Jacoba J E X J v J JXR ER Jw J E J ( ) ( ) ( ) v Basc Jacoba J ( ) J ( ) EX ( ) J( ) v J ( ) osto Repesetatos Catesa Coodates (, yz, ) E ( X) I Cyldcal Coodates (,, z) Usg ( y z) ( cos s z) E cos s ( X) s cos Sphecal Coodates(,, ) Usg ( y z) ( coss ss cos ) E cos s ss cos ( X) s cos s s cos cos s cos s 6

7 Eule Agles Spatal Mechasms s. c c. c s s R ; ER( R) c s s c s s Sgulaty of the epesetato fo k {} {} {} opagato of veloctes v : lea velocty agula velocty {} v he Jacoba (EXLICI FORM) Z Revolute Jot smatc Jot V j V Z he Jacoba (EXLICI FORM) Effecto smatc V j Revolute Lea Vel: V j Agula Vel: oe Effecto Lea Velocty v [ V ( )] Effecto Agula Velocty V j V Z Z v he Jacoba (EXLICI FORM) Effecto smatc V j Revolute Lea Vel: V j Agula Vel: oe Effecto Lea Velocty v [ Z ( Z)] Effecto Agula Velocty ( Z ) V j V Z Z v v [ Z ( Z )] [ Z ( Z ( ) )] Z v Z( Z ) Z ( Z) v J Z Z Z Z Z Z J v 7

8 he Jacoba J J J Mat v F I HG K y J... z Jv v w J v (dect dffeetato) Jacoba a Fame Vecto Repesetato F J G. Z. Z. Z I {} J H F HG I KJ. Z. Z. Z I KJ Stafod Schema Am d d Z Y X6 Y 6 X Z 6 J F HG Z Z Z Z Z Z4 Z5 Z6 I KJ L N M Stafod Schema Am Jacoba J Z Z Z4 Z5 Z6 cd s s d c c d c s sd csd scd ss sd c s c s c c s s c c c c s s s s c s c c s s s c s c c s c c s c s s s s c c s s s c s c c O Q Velocty/Foce Dualty J J F 8

9 Leazed Kematc Model J ( ) Istataeous Ivese Kematcs Resolved Moto-Rate (Whtey 97) J ( ) Jacoba J y Ivese Jacoba J y Redudacy J y Geealzed Ivese J I J J ull space A y System ( m) ( ) ( m) y A ( ) Geeal Soluto RA ( ) m y A I A A y Rage space Colum space of A Geealzed Ivese A( m) ; ak( A) A AA A A : ( m) Eample A ( ) A a a 9

10 A Eample y Ay ( ) y a ( a ) y y a a ( a ) Ay ( ) a A y ( m) ( ) ( m) m Less euatos tha ukows ( m) Fee vaables solutos Eample y y y m ( ) Moe euatos tha ukows At most oe soluto Eample y 4 4y y y 4 4 soluto f s pla spaed by ad 4 4 colum Jacoba Geealzed Ivese colum Geealzed Ivese J : J J J J soluto f s pla spaed by ad 4 4 Geeal Soluto J I J J

11 Geeal Soluto J I J J I J J o I J J J J I J J J JJJ J : J J J J J seudo Ivese A AAA J A AA A A A AA A A AA J : J J Jo J A : uue seudo-ivese Geealzed Ivese Left Ivese m ( ) m Rght Ivese m ( m) A AA A A A I A A A A AA I A A ( AA ) A A I Left Ivese m ( ) m A AW A AW A A I A A A A AA I Rght Ivese m A W A ( AW A ) ( m) AA I

12 Reducto to the Basc Kematc Model Ital oblem ( Reduced oblem ( m J euatos) m EX ( ) J ( ) euatos) J EJ Solvg EX ( ) EX ( ): m m mat ( m m ) ak E X ( ) m ( ) m ak E X at cofguato whee the epesetato s sgula Left Ivese If ak E( X ) m the system has a uue soluto: ( ) E( ) X m m E : s such that E E I m E E E E ad E E ( X) ( Xp) E ( X) p System E m m m m E E E E E E mm m m m m E E E E E E E E E E E Left Ivese I osto Repesetatos Catesa Coodates (, yz, ) E ( X) I Cyldcal Coodates (,, z) Usg ( y z) ( cos s z) E cos s ( X) s cos osto Repesetatos Catesa Coodates (, y, z) E ( X) E ( X) I E Cyldcal Coodates (,, z) cos s ( X) s cos

13 E Sphecal Coodates (,, ) Usg ( y z) ( coss ss cos ) cos s ss cos ( X) s cos s s cos cos s cos s E Sphecal Coodates (,, ) cos s ss cos cos ( X) s s coss s cos cos s Rotato Repesetatos Decto Coses s sˆ ˆ s ; E( ) s s sˆ E E E E ˆ ˆ ˆ ˆ ˆ E ˆ E S S SS SS Eample C S S SSS S C z z z zˆ z z z z z z S Sˆ C S C C Sˆ S C S S SC ˆ ˆ S S SC C SC SC C SC ˆ ˆ SS SC S

14 Sˆ ˆ E E ˆ S S I ˆ S ˆ S X S S ˆ E E ˆ S S I S E E E E E E E E Sˆ Sˆ Sˆ E SSS ˆˆˆ I Agula Velocty s sˆ ˆ s ; E( ) s s ˆ s X E Soluto E X Decto Coses Rotato Eo Istataeous Agula Eo S Sd S ; d S d S S d S Sd S Sd S Sd E X E X S Sd X S Sd S Sd 4

15 E E E Sˆ Sˆ Sˆ E ˆ ˆ ˆ S S S S E ( ˆ ˆ ˆ S SS SS SS) S Istataeous Agula Eo desed SS ˆ SS ˆ SS ˆ d d d Rotato Repesetatos Decto Coses s sˆ ˆ s ; E( ) s s ˆ s Obsevg E E I ˆ ˆ ˆ E S S S E Eule Agles SC CC S S E ( X) C S S C S S cos s s s cos s cos Eule aametes E ( ) 5

16 Eule aametes Obsevg I E Ivese of the Basc Kematc Model System J ( ) ; ( ) m m mo ( ) ( ) Rght Ivese A soluto ff ak J m a m ght vese J J J I m System J ( ) ; ( ) m m mo Redudacy (w..t a ask) Soluto ( ) ( ) J X J : Geealzed Ivese Geeal Soluto J X I JJ lc lc lc y l Sl S l S l l l l S S S J ( ) C C C J J J J l l JJ JJ 6 5 S C S CS C C JJ S det C C C S 6S CS S J () C S 6S (5 ) C S J I J J 5 J J 6 5 6

17 J I J J I J J Redudacy System J ( ) ; ( mo m ) ( m ) ( ) Geeal Soluto J X I JJ Kematc Sgulaty he Effecto Localty loses the ablty to move a decto o to otate about a decto - sgula decto J J J J det det J j J det J Kematc Sgulates lclc y ls ls ls ls ls J lc l S lc det J l l S Sgulaty at k l l 7

18 J S J () C S ls ls J S C l l C l C At Sgulaty J l l l he ak of ( JJ ) ( J J) Sgula Value Decomposto heoem - Defto Ay m mat A of ak ca be factoed to: ; whee U A U V s a m m othogoal mat; V s a othogoal mat; s a m mat of the fom dag (,, ) wth ; wth ae uuely detemed fo A ad called Sgula values of A Decomposto of A A U V ( m) ( mm) ( m) ( ) m A AV V o ) det AA I,, o ) A A I V V o ) AV U Av Av u u u Av U m AA U U ( mm) 8

19 AA o ) det I o ) A A I U U o ) A U V Au Au vv v Au V seudo Ivese of A AA A A A A A A V U A A AA A U V s seudo Ivese of s Eample o )? J () l J J l l l l ll l l l l l l I det J J l l l l l l o ) V? J J Iv l l l l lll ll v ll l V l l l l l l 9

20 o ) U? u Jv u l l (.) l l l (.) (.) (.) l u U J S U V R R l l y () l l l l l l l l l J () () J () () R R l l y () l l y () l l l y () l l l l (.) (.) C S l l l J S C l l l (.) (.) J V U S J l l l l l l ll C S l S C l l l Geeal Soluto J I J J ll l J J l l l l l l

21 l l l l l ll ll l ll J J oblem wth the seudo Ivese Soluto () C S ls ls J S C l l C l C lc ls C S ll S l l C l S S C J() J small J () l l l l ll l R X y y l l J X () () l l y ll l () () () y () y l () ( l l ) () l y() y l () seudo Ivese Soluto l l l l l y ()

22 Sgulaty Robust Ivese Sgulaty Robust Ivese seudo-ivese J J J J S-R Ivese J * J J J ki k k. k k k. k.