1. Introduton easurement and Calbraton of Hgh Auray Spheral Jonts Ale Robertson, Adam Rzepnewsk, Alexander Sloum assahusetts Insttute of Tehnolog Cambrdge, A Hgh auray robot manpulators are requred for numerous ndustral applatons; suh as nlne nspeton of assembly lnes; laser uttng; and preson pk and plae operatons. Parallel knemat mahnes [1] offer benefts, ompared to seral manpulators, due to ther hgh stffness and nherently superor auray. any parallel robot mplementatons utlze two and three degree-of-freedom jonts to onnet the robot arms, wth unversal jonts ommonly used for the two degree-of-freedom jonts. Sgnfant researh has led to pratal mplementatons of three degree-of-freedom jonts based on a modfed unversal jont onept. Whereas ths smplfes jont onstruton, t ntrodues sngulartes and mehanal toleranes that degrade the overall robot auray. The goal of ths researh s to develop a hgh-stffness spheral jont apable of a large, sngularty-free workspae wth auray on the order of several mrometers. Spheral jonts have been developed based on the ball and soket onept, wheren a spheral ball sts n a hemspheral avty. Three desgns are mplemented; based on pont and surfae ontat mehansms. The frst jont (pont ontat) s aheved by knematally mountng the ball on three smaller spheres embedded n the soket, as per ball bar mehansms []. The seond jont (rollng ontat) uses ball transfers nstead of fxed spheres to sgnfantly redue frton. The thrd prototype mplements surfae ontat through replaton of the ball wth a teflon-laed epox to aheve a hgh auray and low-frton ball to soket matng surfae [3]. Eah jont desgn norporates a magnet preload wheren the soket s populated wth varous onfguratons of permanent magnets that attrat the sold steel ball. Eah spheral jont prototype s tested usng a spheral knemat test rg that allows the ndependent atuaton of eah of the jonts degrees of freedom: roll (rotaton about a nomnal x-axs), pth (y- axs rotaton), and yaw (z-axs rotaton). Eah prototype s measured by affxng a toolng ball to a shaft attahed to the ball and mountng the jont n the test rg, whh tself s mounted n a oordnate measurement mahne (C). The test rg atuates the jont to varous postons n ts workng spae and the C measures the poston of the enter of the toolng ball. Jont auray s haraterzed usng a geometr sphere-ft algorthm to ft the measurement data to a standard sphere through the alulaton of the best-ft sphere enter and radus. The fnal (sphere-ft) error s the devaton of eah measurement poston from the best-ft radus. Addtonall a knemat parameter estmaton algorthm has been developed to haraterze the ndvdual error parameters wthn the jont, suh as the eentrty of the axes of rotaton. The passve jont s treated as a three degree-of-freedom mehansm and the error parameters determned through a numeral Jaoban-based tehnque [4]. As the jont prototypes have no angular feedbak and the test rg s relatvely naurate, jont angle postons are not aurately known and needed to be re-alulated for eah measurement loaton. The albraton sequene nvolves the ntal estmaton of knemat error parameters (based on the urrent jont angles) followed by the update of the jont angles (based on the updated error parameters and the measurement loaton). Ths leap-frog algorthm ontnues untl both the error parameters and the jont angles onverge; the fnal (albrated) error s the dfferene between the predted and the measured jont poston. easurements of the rollng ontat prototype jont ndate a maxmum sphere-ft error of twelve mrometers (1µm) and a albrated maxmum error of eght mrometers (8µm). The error of the C s estmated to be two mrometers (µm).. Desgn of Prototype Jonts The spheral jont s requred to have a sngularty-free workng range of 180 and auray of the order of several mrometers. Three prototypes have been mplemented, eah omprsng an alumnum soket mahned from 3.5n stok and a n dameter sold steel ball bearng (Fgure 1). Separaton between ball and soket s prevented by a magnet preload fore suppled by a rng of magnets about the rumferene of the soket. Three prmary ontat mehansms have been nvestgated: pont, rollng and surfae ontat.
The pont ontat prototype loates the ball knematally upon three spheres embedded wthn the soket. Gven mnmal Hertz deformaton due to the relatvely low preload fores, the prmary soure of error for ths type of jont s the auray of the ball (typally 5 mron). The man problem wth ths desgn s the frton nherent n the pont ontat regon. The rollng sphere pont ontat prototype seeks to allevate ths frton problem by utlzng rollng ball transfers mounted n the soket n plae of the sold steel spheres. Ths dramatally redues the frton but potentally nreases the errors due to the addtonal nfluene of the ball transfer auray. The full surfae ontat prototype reates an aurate mountng surfae through replaton wth a Teflon-laed epoxy. Ths provdes a larger ontat regon to mnmze ontat stresses wth mnmal frton due to the Teflon ontat surfae. Despte the potental auray of suh a jont, the frton s stll sgnfant. (a) Fxed sphere pont ontat (b) Rollng sphere pont ontat () Replated full-surfae ontat Fgure 1 Spheral Jont Prototypes 3. Desgn of Spheral Knemat Test Rg The spheral knemat test rg (Fgure ) allows the atuaton of eah of the spheral jont s degrees of freedom for the purpose of measurement n a oordnate measurement mahne (C). It omprses three onentr rngs that are atuated by rado-ontrol servomotors: the x-axs motor s mounted on the base; the y-axs motor s mounted on the x-axs rng; and the z-axs motor s mounted on the y-axs rng. The z-axs motor s onneted to the spheral jont va a velro luth-plate and a toolng ball s mounted on the jont shaft to allow measurements by the C probe. The test rg stffness s desgned to be sgnfantly less than the jont stffness n order to avod deformaton of the jont durng measurements. The test rg an be manually atuated wth a rado ontrol unt and automated va a ommerally avalable ontroller. The test rg s mounted n the C and then run through a set of approxmately 30 poses, dstrbuted evenly about the jont workspae. For eah pose, the toolng ball enter s measured, and the set of jont poses and measured postons ompled for analyss. y-axs rng Z z-axs and luth-plate Y base x-axs rng (a) Test rg and spheral jont (a) Test rg mounted n C Fgure - Spheral knemat test rg
4. odelng of Spheral Jont Eah spheral jont prototype s modeled as a smple three degree-of-freedom mehansm, wth three orthogonal axes: roll (rotaton about the jont x-axs); pth (y-axs rotaton); and yaw (z-axs rotaton). Ths allows eah of the jont prototypes to be modeled and measured n the same fashon and the fnal auray haratersts ompared. Two key models have been developed; a sphere ft model and a knemat model. The sphere ft model assumes all axes nterset at a ommon loaton and fts a sphere enter and radus to the measurement data by mnmzng the resdual error between the predted toolng ball loaton and the measured poston (Fgure 3). The remanng resdual error represents the auray of the jont. R (, Y, Z) ε r ε = R ( x x ) + ( y y ) + ( ) r F ( x, y, z, R) = mn ( ε ) r z z Fgure 3 Spheral jont sphere ft model The knemat model extends the sphere-ft model by ntrodung a nomnal forward knemat model between the measurement system and the toolng ball (Fgure 4). Sx frames are ntrodued to desrbe the jont enter n relaton to the C oordnate frame; the roll, pth and yaw jont angles; and the toolng ball enter n relaton to the jont enter. The spheral jont angles are used as nput to the knemat model to determne the poston of the toolng ball enter. In a smlar error mnmzaton proess as the sphere-ft, the nomnal forward knemat model attempts to ft the jont enter and toolng ball loaton (radus) to the measurement data. Ts, Tbr, Tbp, Tby TCP Tby TCP Tbp Tby Tbr Tbp Ts Tbr Frame desrptons: 1. Tm measurement system frame. Ts soket frame 3. Tbr ball roll frame (θ 1 ) 4. Tbp ball pth frame (θ ) 5. Tby ball yaw frame (θ 3 ) 6. TCP toolng ball frame Tm Tm Ts Nomnal Forward Knemats ( x, z) F( θ, θ θ ) = 1, 3 = Tm Ts Tbr Tbp Tby TCP : Fgure 4 -- Spheral jont strutural loop and knemat model The error forward knemats ntrodues poston and orentaton errors between eah of the nomnal knemat frames (Fgure 5) n order to haraterze the loaton and magntude of the jont errors. A numeral Jaoban matrx s omputed from a sequene of jont postons, ths matrx may be sngular f too many parameters are nluded n the error knemats. Through a proess of elmnaton, the offendng parameters are removed untl the Jaoban s numerally well defned. In addton to the jont enter and toolng ball loaton, sx parameters are found to be sgnfant: three orentaton and three poston errors representng the poston and orentaton of the atual jont enter. A lnear regresson solver, utlzng the sngular value deomposton of the Jaoban matrx, s then used to determne the error parameters by omparng the error forward knemats wth the measured postons. Ths proess s derved from robot albraton applatons [5], however n ths ase the nput jont angles are not well known due to the nauraes of the spheral knemat test rg. Usng a leap-frog approah the jont angles are updated usng a numeral nverse knemats algorthm and the Jaoban and error parameters reomputed untl
onvergene s aheved. If the error parameters are set to zero (only the jont enter and radus are albrated), then the error forward knemats beomes equvalent to both the nomnal knemat and sphere ft models. Error Forward Knemats ( x, z) = F( θ, ε ) Jaoban: J = U S V ( θ ) F ( θ ) Fx 1 x 1 L 1 Fy ( θ1) 1 J = Fz ( θ1) 1 O Fz L 1 Sngular ValueDeomposton : ( θ ) F ( θ ) T : Fx z nn n1 easuremen t : ( x, z) = easured Postons E = F = Error Vetor Calbraton : = parameter update vetor J = E U S V 1 T = V U E S Leap Frog Solver : θ new T = F = E (, θ, ε ) 1 Sgnfant parameters: 1. pth error Ts Tbr. yaw error Ts Tbr 3. y error Tbr Tbp 4. z error Tbr Tbp 5. x error Tbp Tby 6. roll error Tbp Tby Fgure 5 -- Sngular value deomposton and albraton 5. easurement and Calbraton The rollng-sphere spheral jont s measured n a C wth an auray of mron n the workng range of the test rg. The auray dstrbuton s presented n Fgure 6: the sphere ft model results n a maxmum error of approxmately 1 mrometers (µm); the knemat model redues ths error to 8µm. The mean and standard devaton of both models s of the order of 3µm. Ths ndates that the unalbrated jont s aurate to wthn 1µm wthn ts workng range, and ths an be further mproved to 8µm wth the addton of the poston and orentaton error parameters. Resdual Error (mron) 14 1 10 8 6 4 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 0 1 3 4 5 6 7 8 9 30 31 Pose Number Sphere Ft Knemat Sphere Ft ax Knemat ax Fgure 6 -- Auray dstrbuton of rollng-sphere pont ontat prototype 6. Conlusons and Future Work Three spheral jont prototypes have been developed for parallel robot applatons. Eah jont omprses an alumnum soket and a sold steel ball, preloaded by permanent magnets. The pont ontat prototype mounts the ball knematally n the soket. The rollng ontat jont sgnfantly redues frton by knematally mountng the ball on rollng ball-transfers. The surfae ontat prototype uses replaton to aheve a hgh-auray matng surfae. A spheral knemat test rg s used to atuate eah jont to a predefned poston, whh s measured by a
oordnate measurement mahne. The resultant unalbrated auray of the rollng ontat jont was 1 mron, whh s redued to 8 mron through albraton. A fourth prototype s urrently beng nvestgated, based on a flud ontat mehansm. The replated full-surfae desgn s extended to nlude three orfes plaed around the soket to reate a thn flm of ar between the ball and the soket. Ths should sgnfantly redue the frton nherent n the replated prototype whle mantanng the hgh auray expeted of a replated jont. The resdual error dstrbuton remanng after the knemat albraton has a large dsrepany between maxmum and mnmum resduals. Ths ndates that the smple knemat error model nsuffently desrbes the physal phenomena underlyng the measured jont. An mproved model wll be nvestgated wth the goal of ahevng a maxmum error of the order of mron. 7. Referenes [1] Clavel, R. Deve for the movement and postonng of an element n spae. U.S. Patent 4,976,58, 1989. [] Bryan, J. B. Telesopng magnet ball bar test gage. U.S. Patent 4,435,905, 198. [3] Sloum, A.H. Preson ahne Desgn, Soety of anufaturng Engneers, 199. [4] Bernhardht, H. R et al. Robot Calbraton. Chapman and Hall, 1993. [5] Hayat, S., rmran,., Improvng the Absolute postonng Auray of Robot anpulators, Journal of Robot Systems, 1985. Keywords: spheral jont, albraton, metrolog parallel knemat mahnes