On the Error Modeling of a Novel Mobile Hybrid Parallel Robot

Transcription

1 On the Error Moelng of a Novel Moble Hbr Parallel Robot Yongbo Wang (,2), Huapeng Wu (), Hekk Hanroos (), Bngku Chen (2) () Department of Mechancal Engneerng IMVE, Lappeenranta Unverst of echnolog P.O. Box 2, FIN-8 Lappeenranta, Fnlan ongbo.wang@lut.f, huapeng.wu@lut.f, hekk.hanroos@lut.f Abstract hs paper presents a metho for the knematc analss an error moelng of a newl evelope hbr reunant robot IWR (Intersector Welng Robot), whch possesses ten egrees of freeom (DOF) where -DOF n parallel an atonal -DOF n seral. In ths artcle, the problem of knematc moelng an error moelng of the propose IWR robot are scusse. Base on the vector arthmetc metho, the knematc moel an the senstvt moel of the en-effector subject to the structure parameters s erve an analze. he relatons between the pose (poston an orentaton) accurac an manufacturng tolerances, actuaton errors, an connecton errors are formulate. Smulaton s performe to examne the valt an effectveness of the evolutonar algorthm for the applcaton. Kewors accurac, error moelng, parallel robot, knematc analss I. INRODUCION Accurac s an utmost mportant conseraton factor when esgn a robot, whatever t s a seral robot or parallel robot. It s beleve that parallel robot have some favorable avantages, such as hgher spees an acceleratons, compact structure, an mprove accurac because the jont errors are not accumulate lke n ts counterpart. On the other han, seral robots have some avantages lke larger workspace, hgher extert an goo maneuverablt but exhbt low stffness an poor postonng accurac because of ther seral structures. o take avantage both of ther merts, n ths paper, a reunant hbr robot whch possesses both seral an parallel lnks wll be ntrouce, the seral part of the machne s use to prove bg work volume, whle parallel lnks brng hgh loang capabltes an stffness to the whole structure [], thus a promsng compromse of best ses of parallel knematcs an seral robots mght be acheve. In the paper, base on the fferentaton algorthm metho, the error moel of the propose robot wll be formulate. In the past ecaes, there are a number of publcatons concernng the seral robots an parallel robots respectvel. For the Hexapo, Wang an Masor [2] nvestgate how manufacturng an assembl errors affect the accurac of a Hexapo b moelng the legs as seral knematc chans usng the D-H conventon. Ropponen an Ara [] presente an error moel base on fferentaton of the knematcs. For the seral robot, Vetschegger an Ch-haur Wu [] evelope a lnear error moel to etermne the Cartesan poston an orentaton (2) Department of Mechancal Engneerng SLM, Chongqng Unverst Shapngba, Chongqng,, PR. Chna accurac of a robot manpulator wth respect to the statstcal strbutons of the knematcs parameters. However, ver few publcatons ealng wth hbr robot have been foun,.-w Zhao an K.-C Fan [] llustrate a seral-parallel tpe machne tool an evaluate ts accurac base on the lnkage knematc analss an the fferental vector metho. he paper s organze nto four man sectons. he frst secton serves as ntroucton. he secon secton revews the knematc analss an error moelng of the propose robot. Smulaton results are presente n the thr secton, an conclusons are rawn n the fourth secton. II. KINEMAIC ANALYSIS AND ERROR MODELING he knematcs of the propose hbr robot as shown n Fg. can be ve nto two parts, the seral part an the parallel one,.e., the carrage an Hexapo. o smplf ts analss, the two parts wll be frst carre out respectvel, an then combne them together to obtan the fnal solutons. Fgure. D moel of IWR A. error moelng of the carrage Base on the conventon of Denavt-Hartenberg coornate sstem, the prncple of the -DOF carrage mechansm s establshe n Fg.2, whch proves four egrees of freeom at the en-effector, nclung two translatonal movements an two rotatonal movements /8 /$2. 28 IEEE RAM 28

2 p = a sθ a sθ cθ () 2 p = + a cθ + a cθ cθ () z he nverse knematc moel can be obtane as θ = sn p a a x () = p acθ acθ cθ (7) z = p asθ a sθ cθ (8) 2 Fgure 2. Coornate sstem of carrage Usng the coornate sstems establshe n Fg. 2, the corresponng lnk parameters are gven n able. Substtutng the D-H lnk parameters nto (), we can obtan the D-H homogeneous transformaton matrces A, A, 2 A an A. 2 ABLE I. ont D-H PARAMEERS OF CARRIAGE α a θ π /2 a 2 π /2 2 π /2 π /2 a θ π /2 a θ cθ cα sθ sα sθ a cθ sθ cα cθ sα cθ a sθ A = () sα cα where cθ enotes cosθ, an sθ enotes snθ. he resultng homogeneous transformaton matrx can be obtane b multplng the matrces of A, A, 2 A an A. 2 A = A A A A sθ cθ a + + asθ sθ cθ cθ sθ sθ a sθ a sθ cθ = cθcθ sθ cθsθ + acθ + acθcθ Base on (2), the forwar knematcs can be wrtten as follows (2) p = a + + a sθ () x For the accurac of the carrage, t epens on the accurac of the four-lnk parameters of each jont []. If there are errors n the mensonal relatonshps between two consecutve jont an, there wll be a fferental change A between the two jont coornates. herefore, the correct relatonshp between the two successve jont coornates wll be wrtten as A = A + A (9) c where A s the homogeneous matrx whch have the nomnal lnk parameters that can express the relatonshp between the jont coornates an, an A s the fferental change ue to errors n the lnk parameters. It can be approxmate as a lnear functon of four knematcs errors b alor s seres: A A A A A a () = θ α θ a α where θ,, a, an α are small errors n the knematc parameters an the partal ervatves are evaluate wth the nomnal geometrcal lnk parameters. From (), takng the partal ervatve wth respect to θ,, a, an α respectvel, we can obtan A θ sθ cθ cα cθ s α a sθ cθ sθ cα sθ s α acθ =, an A A A,, can be solve n the same wa. a α Let A = A δ A, an A D D D D () δ = θ + + a + α θ a α

3 where D, D, D, D can be solve as follows: a α θ cα sα cα ac α A Dθ = ( A ) =, an θ sα asα D, D, D can be got n the same wa (2) a α Expanng () nto matrx form we can obtan an = [ θ a α ], acoban matrx. G s the entfcaton B. Knematc analss an error moelng of Hexapo Fg. shows a schematc agram of hexapo parallel mechansm, for the purpose of analss, two Cartesan coornate sstems, frames O (X, Y, Z ) an O (X, Y, Z ) are attache to the base plate an the en-effector, respectvel. Sx varable lmbs are connecte wth the base plate b Unversal jonts an the task platform b Sphercal jonts. cα θ sα θ a cα θ α ac α θ + sα δ A = () sα θ α asα θ + cα he above expresson gves the fferental translaton an rotaton vectors for an tpe of jont as functons of the four D- H knematc errors. Smlarl, for the propose four egree-of-freeom carrage, the correct poston an orentaton of the task pont p wth respect to the base frame ue to the knematc errors can be expresse as c = + = + = A A A ( A A ) () Expanng (), an gnorng secon an hgher-orer fferental errors, then the relaton between the fferental change n carrage an the change n lnk parameters can be erve as, ([ ] [ ] ) = A = δa A δa = A δ A A () where δ A s the frst orer error matrx transformaton n the fxe base frame. Followng Paul [7], such a fferental operator has the followng form δθz δθ δ x δθz δθ x δ δ = δθ δθx δ z () If let δ X = δ δ δ δθ δθ δθ R x z x z enote the postonal an the orentaton errors of the carrage, then from () an (), t can also be rewrtten as: ( ) δ X = x = G (7) = = where x = [ δ x δ δ z δθ x δθ δθ z ] Fgure. Normnal moel of the Hexapo parallel mechansm For the esgne knematcs parameters, the followng vector-loop equaton represents the knematcs of the th lmb of the manpulator AB = P + R b a (ι=,2,,,, ) (8) where P enotes the poston vector of the task frame {} wth respect to the base frame {}, an R s the Z-Y-X Euler transformaton matrx expressng the orentaton of the frame {} relatve to the frame {}, cα cβ cα sβsγ sα cγ cα sβcγ + sα sγ sα cβ sα sβsγ cαcλ sαsβcγ cα sγ R = + (9) sβ cβsγ cβcγ an the a, b represent the poston vectors of U-jonts A an S-jonts B n the coornate frames {} an {} respectvel. Let l be the unt vector n the recton of AB, an l represents the magntue of the leg vector AB. Dfferentatng both ses of (8) wll el δll + lδl = δ P + δ R b + R δ b δ a (=,2,..., ) (2)

4 Let R b = s, an multpl both ses of (2) wth the unt recton vector l, snce l l =, l δ l = we can obtan: δl = l δ P + l δ Ω s + l R δ b l δ a ( ) = l δ P + s l Ω + l R δ b l δ a δ P δ a l ( s l ) l l R δ Ω δ b = + (2) Equaton (2) can be rewrtten as δl = δx + δp (22) 2 where [ l, l, l, l, l, l ] δl = δ δ δ δ δ δ R (2) 2 an 2 ( l l R ) l ( s l ) (2) l ( s l ) = R = R ( l l R ) (2) P = a b R ι=,2,..., (2) δ δ δ Snce R s a square matrx, an no sngular ponts exst nse the workspace [], s nvertble. herefore, (22) can be wrtten as: = 2 δx δl δp (27) he frst term on the rght se represents the errors nuce b actuators an the secon one s the poston errors from the passve jonts A an B. C. Knematc analss an error moelng of the hbr manpulator he schematc agram of the reunant hbr manpulator s shown n Fg., whch s a combnaton of carrage an Hexapo manpulator mentone above. he base plate frame {} of Hexapo s conce wth the en task frame of the carrage. he global base frame {} s locate at the left ral. Fgure. Schematc agram of IWR Accorng to the geometr, a vector-loop equaton can be erve as P = P + R P = P + R ( l l + a R b ) = P + R l l + R a R b (28) where P s the poston vector of the task frame {} (or eneffector) wth respect to the fxe base frame {}, an R s the rotaton matrx of the frame {} wth respect to frame {}. Dfferentatng both ses of (28) an multplng unt recton vector l els ( r δ P δ ) = P ( r ) + ( l ) Ω Ω (29) l b l l a l R l l δ δ δ a l R l δl l R l R δ b + + where r b = R b, r a = R a where Equaton (29) can be rewrtten as δx = δx + δ L + δp () ( r l ) = l ( rb l) l b R ()

5 l l R l l = l ( ra l ) + ( l R l l ) ( ra ) + ( l ) R (2) l R l = R l R l ( l R l R ) = R ( l R l R ) () () Fgure. Poston error of carrage n X, Y, an Z Snce R s a square matrx, an no sngular ponts exst nse the workspace, s nvertble. herefore, () can be rewrtten as: δx = δx + δl + δp () where δ X = δ P δ Ω R enote the fnal output pose errors, an the frst term on the rght s the errors cause b the carrage, the secon an thr one represent the errors nuce b the Hexapo machensm. III. SIMULAIONS RESULS In orer to evaluate the fnal output errors cause b the error sources, a smulaton example was performe usng the followng nomnal parameters. a = 28 mm, b = mm, a = 9 mm, a2 =, a = 22 mma, = mm ; = mm, = Moreover, to estmate the accurac of the erve error moel, we assume a certan knematcs errors occurre n the carrage an Hexapo δl =. mm, δp =.mm α = θ =. ; a = =.mm he range of the actuator nput values are gven n below, whch wll be generate b the ranom functon n Matlab. he output poston errors an orentaton errors of the carrage, Hexapo an the whole robot n X, Y an Z recton for the ranom generate poses are shown n Fgure,, 7, 8,9, respectvel. Fgure an Fgure 2 llustrate the comparson of the absolute poston an orentaton error of carrage, Hexapo an the whole robot. Fgure. Orentaton error of carrage n X, Y, an Z Fgure 7. Poston error of Hexapo n X, Y, an Z Fgure 8. Orentaton error of Hexapo n X, Y, an Z < < 8 mm, < < mm, < θ < 8, 2 < θ < 9, < α <, < β <, < γ <.

6 fnal output errors are not smpl the superposton of the carrage an Hexapo. Comparng the absolute poston an orentaton errors of the carrage, Hexapo an IWR, we can see that the carrage error s the most mportant error sources to the fnal output errors, whch causes about 8% of the whole errors. he fnal poston errors are not greater than mm, whch can be reuce to satsf the accurac requrement b means of some calbraton methos n next step. Fgure 9. Poston error of IWR n X, Y, an Z Fgure. Orentaton error of IWR n X, Y, an Z IV. CONCLUSIONS In ths paper, a hbr reunant robot use for both machnng an assemblng of Vacuum Vessel of IER s ntrouce. An error moel erve for the propose robot has the ablt to account for the statc sources of errors. Due to the reunant freeom of the robot, frst we ve t nto seral part an parallel part, an then formulate the error moel respectvel, fnall combne them together to get the fnal error moel. he error moel has been smulate n Matlab an the results show that about 8% amount of errors n the eneffector s cause b seral lnk mechansm,.e. carrage. In practce, to obtan esre accurac of robot, these errors have to be reuce b further parameter entfcaton methos. In the followng work, efforts wll be focuse on the parameter entfcaton usng some optmzaton metho to obtan esrable output errors. ACKNOWLEDGEMENS hs work, supporte b the European communtes uner the contract of assocaton between EURAOM an Fnnsh ekes, was carre out wthn the framework of the European Fuson evelopment Agreement. Authors from SLM n Chna are grateful to the IMVE n Fnlan for the use of ts research facltes an successful collaboraton an Chna Scholarshp Councl for the partal fnancal support. REFERENCES Fgure. Comparson of the absolute poston error of carrage, Hexapo an IWR [] Huapeng Wu, Hekk Hanroos, Pekka Pess, uha Klkk, Lawrence ones, Development an control towars a parallel water hraulc wel/cut for machnng processes n IER vacuum vessel. Int.. fuson Engneerng an Desgn, Vol. 7-79, pp. 2-, 2. [2]. Wang an O. Masor, On the accurac of a Hexapo part I he effect of manufacturng tolerances. IEEE Conf. on Robotcs an Automaton, pp. -2, 99 [].Ropponen..Ara, Accurac Analss of a Mofe Hexapo Manpulator. IEEE Conf. on Robotcs an Automaton, pp. 2-2, 99 [] W.K. Vetschegger, Ch-Haur Wu, Robot analss base on knematcs. IEEE. Robotcs an Automaton, Vol. RA-2,NO., pp. 7-79, september, 98 [] Lung-Wen sa, Robot Analss- the Mechancs of Seral an Parallel Manpulators. Wle & Sons, New York, 2. [].-W Zhao, K.-C Fan,.-H Chang an Z. L, Error analss of a seralparallel tpe machne tool, Int Av Manuf echnol, Vol. 9, pp. 7-79, 22 [7] R.P.Paul, Robot Manpulators:Mathematcs, Programmng, an Control, he MI Press, Cambrge, MA, 982. Fgure 2. Comparson of the absolute orentaton error of carrage, Hexapo an IWR From these Fgures t can be seen that the errors along Z axs are nfluence sgnfcantl than that of X, Y axes, an the