An Operating Precision Analysis Method Considering Multiple Error Sources of Serial Robots

Transcription

1 MAEC Web of Cofereces 35, ( 2015) DOI: / mateccof/ C Owe by the authors, publshe by EDP Sceces, 2015 A Operatg Precso Aalyss Metho Coserg Multple Error Sources of Seral Robots Cog We a, Qgua Ja a Gag Che Bejg Uversty of Posts a elecommucatos, Bejg, Cha Abstract. I orer to solve the problem of operatg precso aalyss coserg multple error sources of seral robots, a operatg precso aalyss metho combe Mote Carlo algorthm wth pose error moel of robot eeffector s propose. Frstly, the pose error moel of robot e-effector s bult base o the MCPC metho. he, the error sources whch have a ma fluece o the operatg precso of robot e-effector are aalyse etal. At last, the Mote Carlo algorthm s use to aalyse the error probablstc characterstcs of robot e-effector coserg multple error sources, whch ca valate the precso esg of mechacal arms a prove theory bass to strbute the error lmts of error sources reasoably. 1 Itroucto he operatg precso s oe of the mportat cators of evaluatg the usablty of robot, a the level of operatg precso wll rectly fluece the effect of task eecuto. he error sources whch affect the operatg precso of the e-effector are varous, such as maufacture a stall error, jot clearace, fleblty of coectg ro a jot, workg evromet a so o[1]. he operatg precso aalyss of the e effector ca be juge f t meets the requremet of precso a prove theory bass to strbute the error lmts of error sources reasoably. here are abuat researches o the precso of the e-effector both at home a abroa the past years. Jeog Kma[2] use the mprove frst-orer secomomet metho to aalyse the effects of the jot clearace o the precso of e effector. Meg Xaju[3] use geometrc moel to stuy the effects of jot moto error o the precso of e effector. a L[4] establshe the mathematcal moel of the fleble jot, whch s use to calculate the pose error of the robot eeffector. Zhag Zhweg[5] researche the effects of coectg ro eformato o the precso of eeffector uer ther ow gravty a loa. I cocluso, although may scholars ha stue the operatg precso of seral robot, the stues whch coserg multple error sources at the same tme for operatg precso are stll rarely. I ths paper, a operatg precso aalyss metho s propose whch combes Mote Carlo algorthm wth pose error moel of robot e-effector. Secto 2 s about the establshmet of pose error moel; Secto 3 s about the aalyss of error sources; Secto 4 s about the a Correspog author: [email protected] probablty aalyss of operatg precso; the cocluso s Secto 5. 2 Pose Error Moel 2.1 Kematc Moel he most classcal kematc moelg metho s base o D-H metho propose by Deavt a HarteBerg[6]. However, whe ajacet jot aes of mapulator are parallel, t wll occur large posto offset o commo ormal le because of the error of parallelsm. herefore, ths paper, the kematc moel s bult by MCPC metho, whch ca esure the tegrty a cotuty of the moel. It uses four parameters,, y, to escrbe the trasformato relato betwee coectg ro coorates, Where, represet the agle rotatg o as a the agle rotatg o y as whe coorate trasformg to coorate 1, respectvely; represets the traslatoal stace betwee the org of coorate a org of coorate 1 the recto of as; y represets the traslatoal stace betwee the org of coorate a org of coorate 1 the recto of y as. A s parameters,, y,,, z s use to escrbe the trasformato relato betwee termate coorate a tool coorate, Where represets the agle rotatg o z as; z represets the traslatoal stace betwee the org of coorate a org of coorate 1 the recto of z as [7]. he trasform matr betwee termeate coectg ro coorates s: Q Rot(, ) Rot(, y ) ras (, y,0) (1) hs s a Ope Access artcle strbute uer the terms of the Creatve Commos Attrbuto Lcese 4.0, whch permts strbuto, a reproucto ay meum, prove the orgal work s properly cte. Artcle avalable at or

2 MAEC Web of Cofereces A the trasform matr betwee termate coorate a tool coorate s: Q Rot(, ) Rot(, y ) Rot(, z ) ras (, y, z ) (2) At last, the kematc moel s erve as follows: 2.2 Pose Error Moel (3) s efe as the fferece betwee the real pose of the e-effector a the esre oe: ( 0,1,, 1) (4) Where, represet the esre trasform matr a the real trasform matr betwee coorate a coorate 1, respectvely. he, the pose error matr ca be erve as follows: 0 z y 1 z 0 y ( ) (5) y 0 z Because there are two etra parameters, z to escrbe the trasformato relato betwee termate coorate a tool coorate, the pose error moel ee am at termeate coectg ro coorate a termate coorate, respectvely. For the termeate coectg ro coorates, s maly relate to the lkage parameters,,, y a the jot parameter : y (6) y Q, Q, Q, Q, Q y, he y ca be erve base o Eq. (4), Eq. (5) a Eq. (6): Q Q Q Q Q y (7) y s efe as postoal error a s efe as atttue error, the the pose error vector D ca be gotte as follows: k k k k k k k k D y (8) For the termate coorate, s maly relate to the jot parameter a the lkage parameters,,, y,, z., y z y z (9) Q, Q, Q, Q Q, Q y, he ca be erve y base o Eq. (4), Eq. (5) a Eq. (9): Q Q Q Q Q Q y Q z y z (10) s efe as postoal error a s efe as atttue error, the the pose error vector D ca be gotte as follows: k k k k k k k D k k k k y z (11), are efe as the esre trasform matr a the real trasform matr betwee termate coorate a ertal coorate, respectvely: ( ) he, s gotte as follows: (12) ( U ) U (13) Where, U E 1 4 E 4 s the etty matr, u u u u u u 1 o a p R p U s erve as follows 0 z y 1 z 0 y ( 1) U U 1 (14) 0 y 0 z he pose error vector D e of the e-effector the termate coorate ca be gotte: p.2

3 l 2 ICMCE 2015 ( ) ( p ) u u u u u u y ( o 1) ( p1 o1) u u u z ( a 1) ( p1 a 1) e u ( 1) u y 13 ( 1) 0 o u z 01 3 ( a1) D (15) he Eq. (15) s euce base o Eq. (8) a Eq. (11): 9 10 M1 M2 M3 k M4 M5 k De 11 M6 M7 M8 k 03 ( 1) 03 ( 1) 03 1 (16) y z J E he Eq. (16) shows the mappg relato betwee lkage errors, jot errors a pose error the termate coorate. Furthermore, the pose error moel of the eeffector D e the ertal coorate ca be gotte: D J J (17) e s E I 3 Aalyss of Error Sources I the part 2, the mappg relato betwee lkage errors, jot errors a pose error ca be gotte the ertal coorate. It wll be coucve to the operatg precso aalyss f all the error sources are trasforme to lkage errors a jot errors Error Aalyss of Maufacture a Istall Error he maufacture error maly shows the legth error L of coectg ro, whch causes the lkage errors, y, z. he stall error maly shows the jot aal evato, whch ca be see Fg.1. AO [, y,0] OO OA AO OA AB AB BC k [ k, ky,0] yae, AE,0 / AE, ta AC (18) Where, eotes the fferetal moto of the jot as; k eotes the rotatoal as; eotes the fferetal agle rotatg o k as; the le AB eotes the real jot as. he, the pose error of correspog coorate ca be gotte as follows: D,,0,,,0 k k ky y (19) he lkage errors k ca be euce base o Eq. (15), Eq. (16) a Eq. (19): Y J D (20) 1 k k k O X Z 2 1 k l 1 Fgure 1. he jot aal evato cause by stall error. 3.2 Error Aalyss of Fleblty of Coectg Ro he force of coectg ro s aalyze, whch ca be see Fg.2. Set the legth of coectg ro as l. he left of the coectg ro s fe, a force F ( F,, ) Fy Fz, torque M ( M,, ) My Mz are apple o the rght of the coectg ro. he uform loa q s create by the coectg ro s ow weght. Because of the effect of forces a torque, the rght of the coectg ro wll move from A1 to A2, a the pose error of correspog coorate ca be gotte as follows: D l wy wz z y (21) Where, l represets the teso or compresso eformato; w y represets the beg eflecto the recto of y as; w z represets the beg eflecto the recto of z as; represets the twst; z represets the agle rotatg o z as; y represets the agle rotatg o y as. he lkage errors ca be euce base o Eq. (15), Eq. (16) a Eq. (21): O 1 Y 1 Z 1 X 1 J D (22) q y 1 q z Y 1 M z O 1 Fgure 2. he eformato of coectg ro. 3.3 Error Aalyss of Fleblty of Jot he fleble jot of mapulator s smplfe to a torsoal sprg, where fleble eformato of the jot s proportoal to the torque of jot[8]: M y F y Z 1 M F z X F p.3

4 MAEC Web of Cofereces C (23) Base o above aalyss, the error sources ca be trasforme to lkage errors or jot errors. Whe the mapulator has a hgh stffess, these varable errors ca be learly ae. he the pose error of the e-effector the ertal coorate ca be gotte through Eq. (17) wth multple error sources affectg the operatg precso of the e-effector. 4 Probablty Aalyss of Operatg Precso he value of error sources has raomess. It s very low probablty that each value of error source has a etreme. If set the etreme of error source as the robot esg cators, t wll crease the cost of maufacture. herefore, the probablty aalyss of operatg precso s eee. I ths paper, the probablty aalyss metho of operatg precso s propose base o Mote Carlo algorthm, a a four lk seral robot s cosere. he 3-D shape of 4-DOF seral robot s show Fg.3. he MCPC Coorates of 4-DOF seral robot s show Fg.4. he MCPC parameters are show ab.1. Lk1 Lk2 Lk3 torsoal rgty of Lk2 s GI p2 4000N m / ra, a the beg rgty of Lk2 s EI Nm / ra ; the qualty of Lk3 s 1.4kg, the torsoal rgty of Lk3 s GI p3 1000N m / ra, a the beg rgty of Lk3 s EI Nm / ra ; the qualty of Lk4 s 3.3kg ; the qualty of the loa s 1kg. he probablty aalyss steps of operatg precso ca be ve to: 1 Obta the probablty strbuto of each error source; 2 Geerate a set of error values of error sources raomly, a trasform them to lkage errors a jot errors. 3 Get the pose error of the e-effector through Eq. (14) usg the varable errors geerate step 2; 4 Perform the steps above N tmes, a the N groups value of pose error ca be gotte. Whe the value of N s bg eough, the probablty strbuto of the pose error of the e-effector ca be gotte. he pose error of robot e-effector s ot oly cocere wth the value of error sources, but also cocere wth the cofgurato of mapulator. Here, the jot agles are , N 2000, the o the probablty aalyss of operatg precso by the steps metoe above. he probablty strbuto of the pose error of the e-effector s show Fg.5 a Fg.6. Lk4 Fgure 3. he 3-D shape of 4-DOF seral robot. Fgure 5. he posto error of the e-effector. Fgure 4. he MCPC Coorates of 4-DOF seral robot. able 1. he MCPC parameters of 4-DOF seral robot. (mm) y (mm) z (mm) \ 0 0 \ \ 0 a1 \ \ a2 0 \ \ a4 0 \ a1a3a5 Where, a0 96, a1 85, a2 1970, a3 96, a4 1770, a5 93 ; the legth error of a 0, a 1, a 3, a 5 s 0.05mm ; the legth error of a2, a 4 s 0.25mm ; the stall error of jot aal s 0.1mm ; the qualty of Lk2 s 1.4kg, the Fgure 6. he atttue error of the e-effector. As show Fg.5 a Fg.6, we ca get the mmum a mamum of the posto error, the mmum a mamum of the atttue error, a the value of the hghest probablty. It ca be use to estmate the operatg precso of the e-effector tutvely, a verfy whether the esg of the mapulator meets the precso requremet. What s more, t ca prove theory bass to strbute the error lmts of error sources reasoably p.4

5 ICMCE Cocluso A operatg precso aalyss metho s propose whch combes Mote Carlo algorthm wth pose error moel of robot e-effector. Frstly, the pose error moel of robot e-effector s bult base o MCPC metho, whch represets the mappg relato betwee lkage errors, jot errors a pose error the termate coorate. he the ma error sources whch fluece the operatg precso of robot e-effector are aalyze etal a trasforme to lkage errors a jot errors. At last, we get the probablty strbuto of the pose error of the e-effector. From the probablty aalyss result of operatg precso we ca verfy whether the esg of the mapulator meets the precso requremet, a prove theory bass to strbute error lmts of error sources reasoably. ACKNOWLEDGMEN hs research s supporte by the Natoal Natural Scece Fouato of Cha ( ) a the Natoal Natural Scece Fouato of Cha ( ) Refereces 1. Jao Guota, Aalyss a sythess of robot pose errors, Post-octor work report of Bejg Uversty of echology. (2002) 2. Jeog Kma, Woo-J Sog, Beom-Soo Kag, Stochastc approach to kematc rela-blty of opeloop mechasm wth mesoal tolerace. Apple Mathematcal Moel-g 34(5): (2009) 3. Meg XaJu, SHI ZhogXu, ZHAN MJg, ZHANG Ce, Aalyss o probablstc characterstcs of moto error lk mechasm. Joural of Mache Desg 20(1): (2003) 4. L a, Research o Fleble Jot Robot a Its Kematc Calbrato a Vbrato Suppresso Reserch. Harb Isttute of echology (2012) 5. Zhag Zhwe, Ga Fagja, Robot Moto Error Base o Lks s Gravty. Mechacal Research & Applcato 01(19): (2006) 6. Ca Zg, Robotcs. Bejg: sghua Uversty Press (2000) 7. Haq Zhuag, Luke K. Wag, Zv S. Roth, Error- Moel-Base Robot Calbrato Usg A Mofue Cpc Moel. Robotcs a Computer-Itegrate Maufacturg 10(4): (1993) 8. Spog M W, Robot yamcs a cotrol. NewYork: Joh wley&sos (1989) p.5