Kinematics, Dynamics & Control

Transcription

1 Kiematics, Dyamics & Cotol DH aametes Ais (i-) Ais i Velocity/oce Duality i- i = i- Li i- z i- i- a i- z i a i i Li i i } i c i -s i 0 a i- s i c i- c i c i- -s i- -s i- i s i s i- c i s i- c i- c i- i J J Reesetatios R Catesia Sheical Cyliical. Eule Agles Diectio Cosies Eule aametes Jacobia fo X Gie a eesetatio Basic Jacobia J ( q) q J q E J q ( ) ( ) ( ) 0 J ( ) 0 q q R

2 Basic Jacobia Jacobia a Basic Jacobia {0} liea elocity agula elocity J J 0 X E J JXR 0 ER Jw I GJ HK (6 ) J ( q) q 0 ( 6 ) ( ) Jq ( ) EX ( ) J( q) J ( ) 0 q q 0 ositio Reesetatios E( ) Catesia Cooiates (, yz, ) E ( ) I3 Cyliical Cooiates (,, z) Usig ( y z) ( cos si z) E cos si 0 ( X) si cos E Sheical Cooiates (,, ) Usig ( y z) ( cossi sisi cos ) cos si sisi cos ( X) si cos 0 si si cos cos si cos si ositio Reesetatios (iese) Catesia Cooiates (, y, z) E ( X) I E E ( ) Cyliical Cooiates (,, z) 3 cos si 0 ( X) si cos E Sheical Cooiates (,, ) cos si sisi cos cos ( X) si si cossi si cos cos 0 si

3 Rotatio Reesetatios Diectio Cosies ˆ ˆ ; E( ) ˆ 3 3 E Diectio Cosies Rotatio Eo Istataeous Agula Eo ; E E Istataeous Agula Eo esie 3 3 ˆ ˆ ˆ 33 E Eule Agles SC CC S S E ( X) C S 0 S C 0 S S 0 cos si si 0 si cos si 0 cos Eule aametes 0 3 E E

4 E Eule aametes E q : Joit Sace Dyamics M ( q): V( q, q ): Gq ( ): : M( q) q V( q, q ) G( q) Geealize Joit Cooiates Mass Mati - Kietic Eegy Mati Cetifugal a Coiolis foces Gaity foces Geealize foces D Cotol Stability Mqq () Bqqq ()[ ] Cqq ()[ ] G () ( qq ) q V / ( qq ) K K V V s ( ) q t q q q q D Cotol Stability Mqq () Bqqq ()[ ] Cqq ()[ ] G () ( qq ) q V / ( qq ) ( qq ) K K ( Vs V ) ( ) s t q q q s q with q 0 fo q 0; 0 s efomace High Gais bette istubace ejectio Gais ae limite by stuctual fleibilities time elays (actuato-sesig) samlig ate es lowest stuctual fleibility elay I lagest elay 3 HG elay KJ samligate 5 Noliea Dyamic Decoulig M( ) V(, ) G( ) M ( ) V (, ) G ( ). ( M M ) M [( V V ) ( G G )] with efect estimates. ( t) : iut of the uit-mass systems ( ) ( ) Close-loo E E E 0 () t 4

5 q q - e - e q Mq ( ) Am q q a f Bqq (, ) Cqq, Gq ( ) as Oiete Cotol Joit Sace Cotol Joit Sace Cotol Oeatioal Sace Cotol Uifie Motio & oce Cotol ( ) V Goal motio V( Goal ) J cotact motio cotact 5

6 Oeatioal Sace Dyamics M V G [ Mˆ, Vˆ, Gˆ, V( )] Goal as-oiete Equatios of Motio {0} { } 0 No-Reuat Maiulato ; m m q qq q : Oeatioal Sace Dyamics M ( ) V(, ) G( ) E-Effecto ositio a Oietatio M ( ): E-Effecto Kietic Eegy Mati V (, ): E-Effecto Cetifugal a Coiolis foces G ( ): E-Effecto Gaity foces : E-Effecto Geealize foces Joit Sace/as Sace Relatioshis Kietic Eegy K (, ) K ( q qq, ) M ( ) qmqq ( ) Usig Jqq ( ) q ( J MJ) q q M q Joit Sace/as Sace Relatioshis E-Effecto Cotol M ( ) J ) ( q) M ( q J ( q) V (, ) J ( q) V( q, q ) M ( q) h( q, q ) G ( ) J ( q) G( q) whee hqq (, ) Jqq ( ) J ( q) 6

7 assie Systems (Stability) V goal g g ( K V) ( K V) System t V ˆ goal V X K K V Coseatie oces goal 0 t Stable Asymtotic Stability a system is asymtotically stable if s 0 ; fo 0 0 s Cotol Gˆ goal K K VV goal 0 s t s Noliea Dyamic Decoulig Moel M ( ) V (, ) G ( ) Cotol Stuctue Mˆ ( ) ' Vˆ (, ) Gˆ ( ) Decoule System I ' with J efect Estimates ' I ' iut of ecoule e-effecto Goal ositio Cotol ' ' ' g Close Loo I ( ) 0 ' ' g Close Loo g g I m0 g ma D Cotol * g Velocity-Lie Cotol * g t 7

8 g * g * with V ma sat if sat sig( ) if 0 ma ajectoy acig ajectoy:,, I ' ' ' ( ) ( ) ' ' ( ) ( ) ( ) 0 o 0 ' ' with I joit sace as-oiete Cotol 0 ' ' q q q with qq q - e - e a f M ( q) J q Am q q af af Ki q Jq V (, ) q q G ( q) 8

9 Comliace I ( ) y 0 0 z ' ' set to zeo ( ) 0 y y ( y y ) 0 ' ' y z z 0 ' Comliace alog Z as Descitio as Secificatio motio foce Selectio mati ; I Uifie Motio & oce Cotol wo ecoule Subsystems * motio * foce 9

10 Natual Systems Coseatie Systems V m 0 System Ietificatio equecy iceases with stiffess a iese mass Natual equecy 0 m () t c cos( t ) t Natual Systems Dissiatie Systems ( KV) ( KV) ( ) f fictio t Viscous fictio: f fictio b m b 0 m ictio Ietificatio System Cotol m f 0 m b f f ( ) Close-Loo m ( b ) ( ) 0 ime Resose 0 m m b m t () ce cos( t) t (t) (t) t t 0