Multi-posture kinematic calibration technique and parameter identification algorithm for articulated arm coordinate measuring machines

Transcription

1 Mult-posture knematc calbraton technque and parameter dentfcaton algorthm for artculated arm coordnate measurng machnes Juan-José AGUILAR, Jorge SANTOLARIA, José-Antono YAGÜE, Ana-Crstna MAJARENA Department of Desgn and Manufacturng Engneerng, Unversty of Zaragoza Centro Poltécnco Superor, c/ María de Luna 3, Zaragoza, 50018, Span. ABSTRACT The dfferent accuracy and repeatablty requrements of artculated arm coordnate measurng machnes (AACMM) and robot make t necessary to consder dfferent dentfcaton technques coverng the characterstc operatonal parameters n each case. Ths paper presents a new data capture technque for subsequent dentfcaton of an AACMM knematc model parameters, usng nomnal data reached by a ball bar and a knematc couplng. Also ths paper addresses the performance evaluaton of the algorthm and objectve functons used, based on a new approach, ncludng terms regardng measurement accuracy and repeatablty. Based on the knematc parameters dentfcaton technque presented, t s possble to develop technques for detectng other AACMM error sources and error correcton of not geometrc sources, such as assembly error, jonts eccentrcty or errors due to temperature changes. A Sterlng seres FARO arm wth 1.5 m long and 6 degrees of freedom (dof) was used to carry out expermental tests n order to evaluate the effcency of the technques presented, showng mproved accuracy and repeatablty performance. Keywords: Knematc calbraton, Artculated arm coordnate measurng machnes (AACMM), Parameter dentfcaton, Data capture, Error correcton, Repeatablty. 1. INTRODUCTION AACMMs make up a specal group wthn coordnate measurement systems due to ther specal characterstcs and dfferences wth regards to tradtonal CMMs. Although ther functon s the same, ther fundamental dfference s the knematc structure. Whle the CMMs, whether they are brdge, gantry or horzontal arm type, have a cartesan confguraton whch allows the measurement of the physcal dsplacement of each of the three lnear axes, the AACMMs employ a seres of rotatng components around generally perpendcular axes. AACMMs adopt the knematc structure and model of robotc arms, and, smlarly, are made up of a seres of straght sectons lnked by rotary jonts whch provde them wth the dof necessary to reach the requred measurement postons. The dfferences compared to robotc arms are great, both n terms of accuracy and functonalty, payng specal attenton to the accuracy of the sensors, materals used and dynamc condtons. Moreover, another mportant dfference compared to CMMs and robotc arms s ther manual and portable operaton, nstead of havng machne axes automatcally controlled. The growng use of the AACMMs has been accompaned by an absence of standards on verfcaton and calbraton procedures, both from the pont of vew of the user and of the manufacturer. Tradtonally, each AACMM manufacturer has adopted ts own evaluaton procedures. Frstly, t s necessary to determne the value of the parameters of the knematc model of the arm. To ths end, each manufacturer uses ts own methods dependng on the model and parameters mplemented n each arm. Both the mathematcal model consdered and the method used to dentfy parameters consttute restrcted nformaton whch s not avalable to the fnal user. Ths nformaton s essental n order to deal wth any accuracy evaluaton of the manufacturer knematc model, although t s not the man goal when establshng general evaluaton methods. In order to fnsh wth ths lack of normalzaton and standardze both the presentaton of results and the methods used n evaluaton, the only standard exstng n the feld of AACMM verfcaton was developed. ASME B , whch sets out the experence accumulated n defnton and carryng out of evaluaton methods of dfferent manufacturers, recommends the methods whch should be followed for relable performance evaluaton of the measurement arms. The metrologcal behavor of an AACMM depends on many factors, although for any commercal AACMM avalable, the fnal measurement error wll depend bascally on the correct determnaton of the mathematcal model parameters [1]. The scant number of bblographcal resources n the feld of AACMM modelng contrasts wth the large number of studes regardng modelng and calbraton of robots and manpulators, from whch t s possble to extract conclusons n order to extrapolate ther applcaton n AACMMs [2-6]. In ths way, despte not fulfllng the condtons of proportonalty and equvalence, the D-H model avods redundances and perfectly descrbes the knematcs of a measurement arm, not presentng ndetermnatons n practcally all AACMMs on the market, whose dual jonts defne consecutve perpendcular axes whch can be thought of as consstng of three jonts, namely shoulder, elbow and wrst. For ths reason, n the present work the AACMM has been modeled by way of D-H when dealng wth the developed dentfcaton procedure, although t s easy to make generally applcable to any knematc model. 2. KINEMATIC MODEL The statc calbraton of an AACMM establsh a parametrc model of ts knematc behavor n order to determne, numercally, the relatonshp between the jont varables and the probe poston for any arm posture. A drect knematc model takes the form of Eq. 1. ( θ, ) y = f q (1)

2 wth =1,,n for an arm wth n rotatng jonts. Ths model calculates the poston and orentaton of the AACMM probe y, accordng to the value of the jont varables θ and to the equatons of the model defned n f, whch depend on the parameters vector q. Ths parameters vector contans the geometrc parameters of the model, whch must be optmzed n order to obtan the lowest possble measurement error. Dependng on the chosen knematc model, the way the equatons are obtaned n f changes, along wth the number of geometrc parameters necessary to be ncluded n q. The D-H basc model uses four parameters (d, a, θ and α ) to model the transformaton of coordnates between successve reference systems. The ntal values taken for the 27 parameters of the D-H model mplemented n the arm (Fg. 1), measured n a CMM or assumed, are shown n table 1. Table 1. Intal D-H parameters n AACMM model posture (Fg. 1). Jont a (mm) α (º) d (mm) θ 0 (º) 1 0,0-90,0 0,0 0,0 2 0,0 90,0 47,8 15,0 3 0,0-90,0 645,83-90,0 4 0,0 90,0 54,5 0,0 5 0,0 90,0 615,7 90,0 6 0,0 0,0 0,0 0,0 Palp (mm) 0,0; Y Palp (mm) 138,5; Z Palp (mm) 54,5 Data capture setup Once the mathematcal model s defned, the next step nvolves the capture of nomnal coordnates n the workspace of the AACMM. All the calbraton procedures, both for robotc arms and AACMMs, establsh a system whch acqures coordnates or nomnal dstances n the workspace, n order to capture ponts whch allow the error to be evaluated and mnmzed. The defnton of postons for dentfcaton n robots ams to cover the maxmum range of jont rotaton, consderng that, n ths way, the nfluences of all the robot arm elements n the workspace are covered. The referenced procedures successfully carry out ther task, and hence both the capture of ponts and the subsequent test postons are captured dscretely n robot confguratons whch are very smlar to each other and also smlar to the common robot work postons. Ths means that the postonng accuracy of the robot s mproved globally n common work confguratons. Snce the optmzaton of the knematc parameters s a least squares method, the adjustment of the parameters whch mnmze the error n the robot dentfcaton postons wll mean small errors occur n postons smlar to those of capture, but wll cause bgger errors n very dfferent postons. In ths work, a contnuous data capture method has been developed. Ths technque allows the massve capture of arm postons correspondng to several ponts of the workspace. To ths end, a ball-bar of 1.5 m long was placed n 7 postons wthn the workspace of the arm n order to cover the maxmum number of possble AACMM postons n a quadrant of the workspace, to subsequently extrapolate the results obtaned throughout ths volume. Fg. 2 shows the consdered postons for the bar. The ball-bar comprses a carbon fber profle and 15 ceramc spheres of 22 mm n dameter, reachng calbrated dstances between the centers wth an uncertanty, n accordance wth ts calbraton certfcate, of ( L)µm, wth L n mm. Fg. 2. Ball bar locatons for each quadrant. Fg. 1. Model defnton posture for the AACMM accordng to D-H conventon. 3. PARAMETER IDENTIFICATION The AACMM used n the present work s a 6 dof Sterlng seres FARO arm wth a typcal confguraton and a-b-c-d-e-f deg rotaton, wth a nomnal value of 2σ = ± mm obtaned n a sngle-pont artculaton performance test of the arm manufacturer, wthout specfyng the number of postons of the knematc seat and ponts captured, and currently out of calbraton accordng to the ASME B sngle-pont artculaton performance test (2σ=0.347 mm) and smple volumetrc performance test (range=0.854 mm). The capture of data both for calbraton and for verfcaton of the arms s usually performed by way of dscrete contact probng of surface ponts of the gauge n order to obtan the center of the spheres from several measurements ponts. In the present work, a specfc probe, capable of drectly probng the center of the spheres of the gauge wthout havng to probe several surface ponts, was desgned. The desgn of the probe s based on general consderatons regardng the desgn of knematc sphere mounts ntroduced n ASME B appendx G. As seen n Fg. 3, the probe comprses three tungsten carbde spheres of 6 mm n dameter, lad out at 120º on the end of the probe. Snce the ceramc spheres of the gauge have a dameter of 22 mm, t s necessary to establsh the geometrcal relatonshps n order to ensure the proper contact of the three spheres and the stablty of ths contact. Thereby, the centerng of the probe drecton wth regards to the sphere center s ensured, makng ths drecton

3 cross t (Fg. 3) for any orentaton of the probe. Thus, t s possble to defne a probe wth zero probe sphere radus and wth the dstance from the poston of the housng to the center of the probed sphere of 22 mm as length, allowng drect probng of the sphere center when the three spheres of the probe and the sphere of the gauge are n contact. Ths confguraton also allows us to measure the ntal value of the three last parameters of the model Probe, Y Probe, Z Probe, to defne the coordnates of the center of the probed sphere wth regards to the last measurement arm reference system. Moreover, ths probe wll be consdered as the reference probe for the later knematc parameter dentfcaton. seres of 400 YZ coordnates measured for each sphere center wll be obtaned. The devatons, ntally due to the value of the parameters of the model between these 400 ponts n each sphere, wll be used to characterze and optmze the arm pont repeatablty. In addton, n each gauge locaton 6 nomnal dstances between the four probed spheres are reached (Fg. 4b). The nomnal dstances of the gauge wll be compared to the dstances measured by the arm. Snce an average of 400 centers per sphere are captured, the mean pont of the set of ponts captured wll be taken as the center of the sphere measured, n order to determne the dstances between spheres probed by the arm (Fg. 4c). Thereby, a method for the subsequent combned optmzaton of the AACMM error n dstances and pont repeatablty s defned. Fg. 3. Inverse knematc couplng probe used n the capture of AACMM postons. Besdes characterzng and optmzng the behavor of the arm wth regards to error n dstances, ts capacty to repeat measurements of a same pont s also tested. Hence, an automatc arm poston capture software has been developed to probe each consdered sphere of the gauge and to replcate the arm behavor n the sngle-pont artculaton performance test, but n ths case, to nclude the postons captured n the optmzaton from the pont of vew of ths repeatablty. The rotaton angle values of the arm jonts for each poston, reached n the contnuous probng of each sphere, are stored to obtan the coordnates of the measured pont wth respect to the global reference system for any set of parameters consdered. In ths way, wth the knematc mount probe n contact wth the sphere, t s possble to capture the maxmum possble number of arm postons, thus coverng a large number of arm confguratons for each sphere consdered. Fg. 4a shows the capture scheme followed. Postons causng maxmum varaton of the arm jonts n all the possble drectons at the start, end and mdpont of each trajectory wll be searched. The capture wll be contnuous and we wll try to capture data n symmetrcal trajectores n the sphere, n order to mnmze the effect of probng force on the gauge. Thereby, around 400 rotaton angle combnatons θ Enc (=1,,6) have been captured for the jonts to cover the postons of the arm probng the center of the measured sphere. Followng ths confguraton, 4 spheres of the gauge n each of the 7 postons consdered for each of the quadrants of the arm work volume were probed, what makes a total of 10,780 angle combnatons. The measurng of a sphere center wth the knematc seat probe from dfferent arm orentatons should result n the same pont measured. The unsutable value of the knematc parameters of the model wll be shown by way of a probng error. Ths error produces dfferent coordnates obtaned for the same measured pont n dfferent arm orentatons. In ths manner, by probng four spheres of each poston of the gauge wth an approxmate average of 400 arm postons per sphere, a Fg. 4. a) Jont angles data capture procedure; b) Dstances between probed spheres centers; c) Center consdered to evaluate dstances between spheres measured and pont repeatablty. Nonlnear Least-Squares Identfcaton Scheme Kovac and Klen present n [7] an dentfcaton method based on nomnal data obtaned wth the gauge developed n [8]. Ths method uses an objectve functon as used n robots, along wth commercal software to dentfy knematc parameters, wthout focusng the study on the partculartes of the measurement arms. In [9], Furutan et al. descrbe an dentfcaton procedure for measurement arms and present an approxmaton to the problem of determnaton of AACMM uncertanty. Ths study s centered on the type of gauge to be used accordng to the arm confguraton and analyses the mnmum number of necessary measurement postons for dentfcaton, as well as the possble gauge confguratons to be used. Agan, ths work does not specfy the procedure to obtan the parameters of the model, nor the type of model mplemented, and does not show expermental results for the method proposed. In [10], Ye et al. develop a smple parameters dentfcaton procedure based on arm postons captured for a specfc pont of the space. As ndcated n secton 2, the knematc model mplemented n the AACMM can be descrbed, for any arm poston, by way of Eq. 2, based on the formulaton of drect knematc problem. ( α θ θ ) p = f a,, d,,, Y, Z, = 1,...,6 (2) 0 Probe Probe Probe Enc n whch p=[ Y Z 1] T are the coordnates of the pont measured wth respect to the arm global reference frame at the base, correspondng to the value of the geometrcal parameters and to the jonts rotaton angles n the current arm poston. There are many alternatves when dealng wth an optmzaton procedure, although the most wdely used n the feld of robot arms and AACMMs are the formulatons based on least squares fttng. Gven the non-lnear nature of the arm knematc model, t s not possble to obtan an analytcal soluton to the problem of parameter dentfcaton. Therefore, t s necessary to use non-lnear optmzaton teratve procedures. In ths way, for the

4 mathematcal formulaton of the optmzaton method t s common to defne the objectve functon to mnmze n terms of square error components. Based on the nomnal coordnates reached by the gauge and those correspondng to the ponts measured, we can obtan the arm measurement error as the Eucldean dstance between both ponts. Snce the dentfcaton procedure both n robots and n AACMMs s based on the capture of dscrete postons wthn the workspace, all the revewed optmzaton procedures use Eq. 3 as basc objectve functon to mnmze. m T φ = [ Δp] [ Δp] = 1 (3) T [ Δ p] = [ δx δy δz] = p p ( ) 0 Eq. 3 quantfes the error n dstances between the nomnal pont and the pont reached for all the postons captured, formulated as the quadratc sum. In ths work, n order to choose the objectve functon to be mnmzed, consderaton has been gven to the error n dstances for the 42 dstances measured. Therefore, t s possble to evaluate the 10,780 combnatons of sx values of jont angles for each set of knematc parameters, and to obtan the centers as the mean value of the coordnates correspondng to each sphere. Fnally, we evaluate all the dstances n each teraton of the optmzaton procedure. Hence an objectve functon smlar to those commonly chosen n robot and AACMMs parameter dentfcaton s obtaned. Gven the arm postons capture setup used, and the fact that pont repeatablty n any arm probe orentaton s a very mportant parameter n order to characterze the metrologcal behavor, we use a new objectve functon n the optmzaton. Unlke tradtonal expressons, our objectve functon (Eq. 4) ncludes both the errors n dstance and the devaton of the ponts measured n each sphere showng the nfluence of the volumetrc accuracy and pont repeatablty, mnmzng smultaneously the errors correspondng to both contrbutons. ( D D0 ) ( 2 ) ( 2 ) ( 2 ) jk jk j Yj Zj r s = 1 j, k= 1 (4) φ = + σ + σ + σ In the objectve functon proposed, wth the capture setup descrbed, r=7 postons of the ball bar and s=4 spheres (1, 6, 10 and 14) per bar poston. Agan, n Eq. 4 t s necessary to consder the elmnaton of the terms n whch j=k, n order to avod the ncluson of null terms or consderng as duplcate the nfluence of the error on dstances, takng nto account that D = D jk. The frst term of Eq. 4 corresponds to the error n kj dstances n poston of the gauge between sphere j and sphere k, whereas the other terms refer to twce the standard devaton n each of the three coordnates for sphere j n poston of the gauge. Fnally, agan by mathematcal formulaton of the optmzaton problem, t s necessary to consder the sum of all the square errors calculated. Usng the objectve functon of Eq. 4, 126 quadratc error terms wll be obtaned to calculate the fnal value of the objectve functon after each optmzaton algorthm stage. Ths value wll show the nfluence of the knematc parameters as well as of the jont varables through the calculaton of the ponts coordnates correspondng to the 10,780 arm postons. The Levenberg-Marquardt (L-M) method [11, 12] has been chosen as optmzaton algorthm for parameter dentfcaton, gven ts proven effcency n robot parameter dentfcaton procedures. The selecton of a specfc optmzaton procedure mples to avod the nfluence of the mathematcal method tself wth regards to the data captured on the result. One of the most sutable methods to solve ths problem s the L-M algorthm. Table 2 shows the AACMM knematc model parameters fnally dentfed, based on the ntal values of Table 1 for the objectve functon of Eq. 4 and the 10,780 arm postons consdered. The error values obtaned for the dentfed set of parameters are shown n Table 3. Table 2. Identfed values for the model parameters by L-M algorthm. Jont a (mm) α (º) d (mm) θ 0 (º) Probe Y Probe Z Probe (mm) (mm) (mm) Table 3. Error ndcators for the dentfed set of model parameters. Dstance error (mm) 2σ by sphere (mm) Max Max Causng Pos. POS2 Causng Pos. POS 1 Causng Dst. D1 Causng Sph. B1 Mn Causng Coord. Z Causng Pos. POS1 Mn Causng Dst. D2 Causng Pos. POS4 Mean Causng Sph. B6 Causng Coord. Y Mean In order to study the nfluence of the ncluson of the standard devaton on the objectve functon, we have complete optmzatons takng as functon only the terms correspondng to the error n dstances for the 10,780 postons captured, as would correspond to a common objectve functon for parameter dentfcaton of robots. Compared to the maxmum and mean error obtaned n Table 3, n ths case a maxmum error of 15 µm was obtaned and a mean error of 5 µm for the same arm postons. However, for the parameters dentfed wth the objectve functon of Eq. 4, the maxmum value obtaned for 2σ s mm compared to mm obtaned usng only dstance terms, and the mean value s mm. As can be seen n the results, an optmzaton equvalent to those commonly found n robots produces excellent results for errors n dstance but nadequate results for range and standard devaton. Hence, to obtan a set of parameters whch allows the arm to be repeatable n a pont for any measurement orentaton and not only n the orentaton captured for optmzaton, t s necessary to consder the range or the standard devaton n objectve functon. 4. TEST AND RESULTS The generalzaton of an dentfed set of parameters to all the measurement volume nvolves the obtanng of devaton and error values smaller than the maxmums obtaned for the

5 dentfcaton process for any arm poston. Once the optmzaton process s complete, as the fnal stage of the presented parameter dentfcaton procedure, t s necessary to evaluate the behavor of the arm wth the optmum set of parameters on arm postons dfferent to those used durng dentfcaton. The more smlar the evaluaton postons subsequent to those used n dentfcaton, the better the results. Hence, t s necessary to fnd dfferent measurement arm postons to evaluate the level of fulfllment of the error values obtaned n other measurement volume postons. In ths case, as test bar locaton subsequent to dentfcaton was chosen n the upper part of quadrant 1. Based on the same orentaton of poston P1, the bar was rotated approxmately 25º both horzontally and vertcally. For ths ball bar locaton, angle combnatons correspondng to the arm postons probng the centers of the 14 gauge spheres were captured. In ths way 6,195 arm postons were captured for the test poston, whch s a relable check of the measurement arm error on postons not used. Table 4 shows the error values obtaned for the 14 test poston spheres. Table 4. Error ndcators for the dentfed set of model parameters for ball bar test locaton. Dstance Error (mm) 2σ by Sphere (mm) Max Max Causng Pos. POS_TEST Causng Pos. POS_TEST Causng Dst. D1-10 Causng Sph. B14 Mn Causng Coord. Z Causng Pos. POS_TEST Mn Causng Dst. D1-2 Causng Pos. POS_TEST Mean Causng Sph. B3 Causng Coord. Mean 0,0541 Mean Y 0,0696 Mean Z 0,1411 dstance of mm), t s possble to consder the acceptance of the knematc parameters obtaned, takng them as optmum, and to model the behavor of the ponts devaton based on the arm poston. In ths manner, we obtan a correcton mechansm separate from parameter optmzaton, whch wll not have subsequent nfluences on the error n dstances shown for the optmzed knematc parameters. Gven the nature of the dentfcaton procedure proposed, n whch the mean values of all the ponts captured n each sphere n terms of error n dstances of the objectve functon are optmzed, the correcton model should only be applcable to the error of each pont wth regards to ths optmzed mean, so that no nfluence s ntroduced on the measurement accuracy n dstances, drectly attrbutable to the knematc parameters obtaned. Thereby, we can consder the dfference between each pont and the mean pont of the correspondng sphere - value whch causes the standard devatons obtaned - as an error due to the other nfluences, remanng separate from the knematc parameters optmzaton. It can be assumed that the solated nfluence of each jont s perodcal wth the jont rotaton. Consderng the repeatablty error values descrbed above, we can obtan a functon of 6 varables, one for each jont angle, and try to approxmate ths functon by a Fourer seres. Based on the optmum parameters of Table 2, t s possble to reconstruct all the ponts captured durng the dentfcaton stage and represent the errors n, Y, Z of each one of the 10,780 ponts, correspondng to the 7 postons of the gauge bar wth regards to the mean obtaned for each sphere. If ths error s represented consecutvely for each poston captured n dentfcaton phase, a functon wll be obtaned (Fg. 5), where the repeatablty error of each pont s also defned ndependently for each coordnate. As the concluson of the evaluaton test, the mportance of the data captured should be agan emphaszed. A hgh number of arm postons, dfferent to those chosen for dentfcaton, should be searched n the way recommended n normalzed evaluaton test, n order to conclude wth the acceptance of the dentfed model parameters. In ths case, the number of arm postons consdered for evaluaton s hgh compared to those used n dentfcaton, obtanng values below the maxmum error, meanng the arm behavor s verfed n accordance wth these maxmum errors wthn the volume consdered. 5. NON GEOMETRIC ERROR CORRECTION Besdes the postonng or measurement error due to knematc parameters, whch s n general the bggest source of error, there are other error sources whch nfluence the fnal accuracy of the arm. Thermal, vbraton, electrcal or jont complance errors can be consdered. The optmzaton carred out tres to numercally adjust the parameters n order to obtan the small error as possble, not beng possble to solate and characterze the nfluences of knematc parameters on ths fnal error. Absorbng the error sources means that s not possble to separate and optmze each of them, resultng n a set of parameters whch mnmzes all errors. Also consderng that the optmzatons obtan a sutable pont repeatablty value but not too low compared to the reducton of error n dstances (2σ around 0.4 mm wth maxmum errors n Fg. 5. Pont repeatablty errors for the optmal set of model parameters over dentfcaton AACMM postons. Ths functon can be approxmated by a Fourer seres, whch decomposes a perodc functon nto a sum of smple oscllatng functons, namely snes and cosnes, for each of the jont angles wth coeffcents to be determned by applyng a non-lnear regresson method. Eq. 5 shows the approxmaton functon used n regresson for coordnate, wth analogue functons for Y and Z.

6 ( ( ) ( )) 6 t (, x, x ) = + jk cos j + jk j j= 1 k= 1 fε θ a b C a k θ b sn k θ (5) Regresson tests have been carred out consderng the fundamental harmonc (t=1), 2, 3 and 4 harmoncs. Values of the same order as the repeatablty error were obtaned for resduals for all the tests, mprovng sgnfcantly when ncreasng the number of harmoncs. The best result consderng cross products n Eq. 5 s shown n Fg. 6. Ths result was obtaned for Fourer seres of 2 harmoncs (t=2) plus the cross terms of the frst harmonc, plus the cross terms of the second harmonc, plus the mxed cross terms, resultng n a dependent functon of 219 parameters. Takng the approxmaton functon fnally obtaned for the error model, the evaluaton on the test poston s shown n Fg. 7. Agan, for ths fgure, the real repeatablty error occurrng n the test poston wth the optmum knematc parameters and the estmated error n each coordnate can be seen. 6. CONCLUSIONS Ths work has presented a procedure for knematc calbraton and error correcton for AACMMs. A new method for the capture of data for dentfcaton of AACMM knematc parameters has been proposed and also a new approxmaton to the AACMM knematc parameter dentfcaton problem. Ths approxmaton s based on an objectve functon ncludng terms of error n dstances and terms of standard devaton whch consder the nfluence of arm repeatablty, gven ts capacty to probe the same pont from dfferent orentatons. Fnally, a dynamc correcton model for repeatablty error has been proposed. By way of expermental tests t has been checked that ths model does not affect accuracy n dstances obtaned wth the parameters dentfed and mproves arm repeatablty by more than 50%. 7. REFERENCES Fg. 6. Estmated and real repeatablty error over dentfcaton postons trough Eq. 5 wth 2 harmoncs and cross terms. [1] R.P. Judd, A.B. Knasnsk, Technque to calbrate ndustral robots wth expermental verfcaton, IEEE Transactons on Robotcs and Automaton 6(1) (1990) [2] J. Denavt, R.S. Hartenberg, A knematc notaton for lower-par mechansms based on matrces, Journal of Appled Mechancs, Transactons of the ASME 77 (1955) [3] S. Hayat, M. Mrmran, Improvng the absolute postonng accuracy of robot manpulators, Journal of Robotc Systems 2(4) (1985) [4] T.W. Hsu, L.J. Everett, Identfcaton of the knematc parameters of a robot manpulator for postonal accuracy mprovement, Computers n Engneerng, Proceedngs of the Internatonal Computers n Engneerng Conference and exhbton (1) (1985) [5] H.W. Stone, A.C. Sanderson, C.F. Neumann, Arm sgnature dentfcaton, IEEE Internatonal Conference On Robotcs And Automaton 1 (1986) [6] J.M. Hollerbach, C.W. Wampler, The calbraton ndex and taxonomy for robot knematc calbraton methods, Internatonal Journal of Robotcs Research 15(6) (1996) [7] I. Kovac, A. Klen, Apparatus and a procedure to calbrate coordnate measurng arms, Journal of Mechancal Engneerng 48(1) (2002) [8] I. Kovac, A. Frank A., Testng and calbraton of coordnate measurng arms, Precson Engneerng 25(2) (2001) [9] R. Furutan, K. Shmojma, K. Takamasu, Parameter calbraton for non-cartesan CMM, VDI Berchte 1860 (2004) [10] D. Ye, R.S. Che, Q.C. Huang, Calbraton for knematcs parameters of artculated CMM, Proceedngs of the Second Internatonal Symposum on Instrumentaton Scence and Technology 3 (2002) 3/145-3/149. [11] K. Levenberg, A method for the soluton of certan non-lnear problems n least squares, Quarterly of Appled Mathematcs-Notes 2(2) (1944) [12] D.W. Marquardt, An algorthm for least-squares estmaton of nonlnear parameters, Journal of the Socety for Industral and Appled Mathematcs 11(2) (1963) Fg 7. Predcted and real repeatablty error n ball bar test locaton.