4 Application to the TA-40 Manipulator Introduction Description of the Manipulator

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1 Applitio to the TA-0 Mipultor.. Itrodutio Util ow ol the theoretil priiple ued i thi thei hve ee overed. Thi hpter over how thi theor i pplied to the TA-0 mipultor. The TA-0 i rooti mipultor ued PETROBRAS i uderwter itervetio. It i tthed to ROV (Remote Opertig Vehile) tht will te it to it worig eviromet t gret depth off-hore. The mipultor i urretl otrolled tele-opertio d it doe ot offer the repetilit or olute preiio required to perform more refied utomted t. Firt, rief deriptio of the mipultor will e give, d the more thorough deriptio of ever li d joit tht otitute the mipultor. The omil meuremet of the TA-0 will e implemeted i the Devit- Hrteerg ottio to etimte the iemti of the mipultor... Deriptio of the Mipultor The TA-0 i hdruli mipultor ple of liftig 0g t the mimum reh of 90mm. It h rottiol joit, reultig i degree of freedom. At the ed-effetor gripper i tthed. It h ee reted to operte i hotile eviromet d it i ple of worig t e depth of 000 meter. At preet it i operted mter-lve ofigurtio, where the mter i repreeted miiture mipultor, how i Figure.

9 Figure TA0 d the miiture root ued mter With the ireed preiio d repetilit ttied lirtio, the trjetorie of the root e developed offlie i virtul eviromet, reduig the time d ot of the proe.

2 9 Figure TA0 d the miiture root ued mter With the ireed preiio d repetilit ttied lirtio, the trjetorie of the root e developed offlie i virtul eviromet, reduig the time d ot of the proe... Kiemti of the TA-0 A iemti model of the mipultor i eer to perform the lirtio of the mipultor truture. The theoretil prt i dedued i Chpter. Figure how the mipultor d the 7 frme (oordite tem), oe t eh joit d oe t the ed effetor. The followig etio how how the Devit-Hrteerg prmeter of the TA-0 re otied.

3 Figure TA-0 d oordite tem [] 9

4 9... Joit d The eter of joit (O 0 ) i ituted t the mipultor e. The i 0 repreet the rottio i of joit. The i 0 i the ommo orml etwee the frme eter O 0 of joit d O of joit. The fied dite etwee the eter O 0 d O log the ommo orml i mm d it i repreeted i the DH-ottio. Looig i the diretio of 0, the i i rotted 90º reltive to the 0 i. Thi gle i repreeted α 90º. The dite etwee the frme eter i diretio 0 i ero, d it i repreeted d Joit d The dite etwee the frme eter, O d O, log the ommo orml i 7mm givig 7. The rottio e, d, re prllel, givig α 0º. The dite etwee O d O log i ero, givig d Joit d The dite etwee the frme eter i 88 mm givig 88. The poitio of O i outide the truture of the mipultor. The i i rotted 90º roud the i, givig α 90º. The dite etwee the repetive frme eter log i ero, givig d Joit d The frme eter O of joit i loted 77mm log the i from O, givig d 77. Sie the frme eter poitio log the ommo orml i ero, 0. The i i rotted -90º reltive to, givig α -90º.

5 9... Joit d The frme eter, O d O, re ituted t the me poitio, givig d 0, 0. The i i rotted 90º reltive to the i, givig α 90º.... Joit O i ituted 0mm log the i. Thi give d 0 d 0. Sie there i o joit loted t O, the oriettio of frme e hoe ritrril log d re prllel whe θ 0. The i i hoe o tht it oiide with the i. There i o rottio log the ommo orml givig α 0º...7. Devit-Hrteerg Prmeter Tle oti ll the Devit-Hrteerg prmeter. From thee the prmeter trformtio mtrie, Ai, e give to lulte the iemti of the mipultor uig Eq. () d (8). Li i i [mm] d i [mm] α i [ o ] θ i 0 90 θ θ θ θ θ θ Tle Devit-Hrteerg prmeter

6 9.. Clirtio of the TA-0 Thi hpter epli how the theor i hpter. i pplied to the TA-0 mipultor. I order to etimte the geerlied error, ll the redudt error hve to e elimited. Thi i doe trferrig the vlue of the redudt error uig Eq. () for i:. The redudt error ε,(i) d ε r,(i) ould the e elimited for i0:. The error ε,() ot e elimited ie it otriutio to the ed-effetor poitio i ot ped o to other joit. The error ε p, ε,, d ε r, e elimited ie the re rottiol error d do ot effet the ed-effetor poitio. Thi elimite error i totl. Further, there eit other reltio etwee the redudt error give i Eq.(7). Rerrgig thi equtio give: ε ε ε d ε *,() εp,() εp,() d *,(),(),() (7) Thi me tht ε,, d ε, e elimited from the model, levig ol error to etimte. The redued idetifitio joi (G e ) h ol elemet. The mtri GG i the ivertile. Sutitutig J t with G e i Eq.() T e give olutio to the equtio. e Elimitig ε, d oitig it with the trltiol error me tht the oriettio error of the ed-effetor ot e etimted idepedetl. Neither the oriettio of li. However, the etimted oriettio of li will hve fied i to the true oriettio for ll ofigurtio of the joit. Thi me tht mer o li will e le to detet the poe differee etwee two view.

7 9.. Ivere Kiemti The ivere iemti i thi hpter w deduted i []. It i preeted i thi thei due to the importe for utomtio purpoe. It i impoile to develop geerl method to etimte the ivere iemti for mipultor. Therefore the tep developed i thi hpter ot e pplied diretl to other mipultor. Uig the peifi propertie of the TA-0 me it poile to fid olutio. The joit,,, d re ll ituted withi ple i D pe. Their repetive frme (oordite tem) re O, O, O, O d O. A grphi iterprettio of thi ple i give i Figure. Equtio (8) give the poitio d oriettio of the ed-effetor i e oordite. T ( ) θ A A A A A A (8) 0 0 Equtio (8) e elorted to give the oordite of frme, P, reltive to frme : ( ) ( ) 0 0 P P A T A A A A A (9) Equtio () give the reltive poitio d oriettio of frme reltive to frme. The frme d hve the me poitio, meig tht the gle of joit doe ot ffet the poitio of frme. B iterprettio of Figure the followig equtio re otied: d R d A A A A P (70) 0 0 ( A ) P ( A) ( ) ( ) ( ) ( ) 0 0 R (7)

8 97 ( ) ( ) ( ) ( ) ' ' ' 0 d d P P (7) From the third lie i Eq.(7) the gle of the firt joit i otied: θ π t (7) Uig the firt two lie of Eq.(7) give: ' ' d d (7) The movemet etwee frme d frme e iterpreted mipultor with degree of freedom, ie the dite () etwee O d O i ott. Figure - A D iterprettio of the frme O, O d O. Frme O oiide with frme O []. From Eq.(7) d the D iterprettio of the geometr i Figure the followig equtio re otied: φ θ ϖ ϖ θ ϖ θ ' ).i(.. ' ) o(.. d d d (7)

9 98 Solvig the equtio for θ d θ give: ' θ t o φ ' (7) θ o t d ' (77) Hvig the gle of the firt three joit d the deired poitio of the ed-effetor, it i poile to etimte the required gle of joit d. The movemet of joit doe ot hge the poitio of the ed-effetor, ol the oriettio. The movemet etwee frme d the ed-effetor i give Eq.(78). d P AAA R d d d 0 '' 0 0 '' P ( A ) T ( A AA ) '' (78) (79) 0 ( ) R A T (80) 0 ( ) ( - ) ( - )( )- - P - ( )( - )( )( )- '' d (8) '' d '' d d All the vlue i Eq.(8) re ott ie the gle of joit to hve lred ee foud. From Eq. (8) the gle of the joit d re otied:

10 99 θ π '' '' t (8) ( ) '' t '' d θ (8) Whe the gle of the firt five joit re otied, the poitio of the ed effetor i lred determied. Joit i ol hge the poitio of the edeffetor. Therefore the gle of the ith joit i otied from the deired oriettio of the ed effetor: p p p A (8) ( ) [ ] ( ) ( ) [ ] ( ) ( ) (8) To implif the equtio, the followig vrile re itrodued: ( ) [ ] ( ) ( ) [ ] ( ) ( ) From the gle of the joit to : (8) Solvig Eq.(8) give: t θ π (87) The ivere iemti equtio oti m trigoometri term whih etil m poile olutio for deired poitio of the ed-effetor.

11 00 Equtio (7) h two olutio. Due to phil limittio, the gle of joit eed to e etwee -0º 90º. Equtio (7) d (77) tht refer to gle θ d θ repetivel hve two poile olutio eh. Kowig tht joit ol tti poitive gle elimite thi miguit, llowig ol olutio tht give poitive gle for thee joit. Equtio (8), whih give the gle of joit, etil igulrit whe the gle of joit i ero. I thi e the joit d re redudt. Joit the h to e fied i ritrr poitio. Joit i the djuted to give the deired oriettio of the ed-effetor.

12 0.. Oriettio Error of the Mipultor Aordig to the model of geerlied error, the et trltio d oriettio of the mipultor ed-effetor iludig error i give Eq.(0). Whe the mipultor truture h ee lirted, the ed effetor poitio e etimted. However, the oriettio of the ed-effetor ot e etimted urtel ie the rotr error ε, h ee elimited d trferred to the trltiol error ε,. Alo, the trltiol error ε, h ee elimited d trferred to the rotr error ε p,. Thi give the right poitio, ut the trltio error of the ed effetor i ot poile to etimte uig the redued et of error. If the mer i pled o li of the mipultor, it i the rottio error of li tht eed to e oidered. The homogeeou mtri tht derie the oriettio d poitio of the li reltive to it e futio of the gle of the joit θ[θ θ θ θ θ θ ] d the geerlied error ε i give : T ( θε, ) E A E A E A E A E A (88) Thi equtio give the tul poitio d oriettio of li give full et of geerlied error. After the lirtio of the root, ol the idepedet uet, ε i ville. Thi me tht the elimited error of ε hve to e utituted ero i the geerlied error mtrie. The devitio etwee the true d etimted oriettio d poitio the e give : 0 X 0 0 0, 0 R Y T T ( θε, ) T ( θε, ) 0 Z (89) The ture of the devitio e viulied i imultio. B firt defiig full error vetor d the etimtig the redued et of error, the effet e imulted.

13 0 o θ 0 i θ o i θ 0 θ R 0 0 i θ o θ 0 0 o θ i θ i θ 0 o θ i o θ 0 θ R R R R R R R R R The fil epreio for the reltive rottio mtri will e: (90) R o θ o θ i θ i θ i θ R i θ o θ R i θ o θ i o i θ θ θ R o θ i i i o θ θ θ θ R o θ o θ R i θ o θ i o o θ θ θ R i θ o θ R i θ R o θ o θ From Eq.(9) the reltive rottio gle e determied. (9) R θ t (9) R (,) (9) θ i R R θ t R (9) To get ide of the mgitude of the oriettio error of li there imultio w performed. The tul geerlied poitio error were hoe rdoml i the itervl /- mm d the rottiol error i the itervl /- /- π/80 rdi. 00 meuremet of the ed-effetor were imulted d the redued error vetor, ε, w etimted.

14 0 Error ε Atul ε Etimted ε ε,0 -, , ,7 0 - ε.0-7, , ,7 0 - ε.0, ε.0-7, , ,7 0 - ε r.0-7, ε p.0 -, , , ε. -, , , ε.,09 0 0, 0 0, 0 0 ε. -, ε. -, , , ε r. -, ε p.,0 0 -,0 0 -,0 0 - ε., , , ε., , , ε. -, ε. -, , , ε r. -, ε p.,70 0 -,70 0 -, ε.,09 0 0,09 0 0, ε., , , ε. -, ε., , , ε r. -, ε p., , , ε. -, , ,7 0 0 ε., , , ε., ε.,78 0 -,98 0 -, ε r., ε p. 9,00 0-9,00 0-9, ε. -, , , ε., ε., ε., ε r. -, ε p. -, ε , ε , ε. 0, , ε ε r ε p Tle Error from imultio Tle how the error tht were ued i the imultio. The differee etwee the tul ε d the etimted ε w le th 0 - for ll elemet of ε.

15 0 The tul mipultor w ot ued i thi eperimet, o the reult ol demotrte the ur of the mipultor give tht the orepetitive error of the mipultor re egletle. The mgitude of the rottio error, R w the etimted for 00 differet ofigurtio of the joit. The ofigurtio of the i joit were hoe rdoml withi the poile movemet for eh joit. The rottio d poitio error for li d the ed-effetor re the plotted. Figure how the poitio error of the ed-effetor for the differet ofigurtio. The grph how tht the poitio error i egligile for uh ig mipultor. Figure - Poitio error of ed-effetor fter lirtio

16 0 Figure 7 how the rottio error of the ed-effetor fter lirtio. It i oviou tht the rottio error of the ed-effetor i too ig to e ued e for the mer. Figure 7 - Rottio error t the ed effetor fter lirtio

17 0 Figure 8 how the poitio error of li. The poitio error i lmot ott. Thi me tht mer tht i tthed li will hve fied devitio from it etimted poitio. Thi devitio e foud through lirtio. Figure 8 - Poitio error of li fter lirtio

18 07 Figure 9 how the rottio error of li. Alo the rottio error i ott. The rottio error roud the i i lrge ompred to the other rottio error. From tle, it e ee tht the error ε. i dl etimted. Aordig to Eq.(7) thi error i oeted with the rottio error roud the i of joit, ε.. Sie the otriutio o the ed-effetor poitio from thee two error ot e ditiguihed, the rottio error roud the i i lrge whe, ε, i lrge. Figure 9 - Rottio error of li fter lirtio After etimtig ll error prmeter, the iemti model of the mipultor e ued to lirte the root e uig the viio tehique previoul deried. Eperimetl reult re preeted et.

Addendum. Addendum. Vector Review. Department of Computer Science and Engineering 1-1

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