Measurement and compensation of motion errors on 5-axis machine tool by R-test Chiaki OYAMA 1, Soichi IBARAKI 1 and Hisashi OTSUBO 2 1

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1 Proceedngs of 4th IRP Internatonal onference on Hgh Performance uttng, 2 Measurement and comensaton of moton errors on 5-axs machne tool by R-test hak OYAMA, Soch IARAKI and Hsash OTSUO 2 Machnng, Measurement and ontrol Lab., Det. of Mcro-Engneerng, Graduate school of Engneerng, Kyoto Unversty, Jaan, c.oyama@at2.ecs.kyoto-u.ac. 2 Otsubo Engneerng Research enter, Jaan Abstract: 5-axs machne tool ntended n ths study has three lnear axes and addtonal to rotary axes. ecause of ts comoston, errors of each axs and assembly errors are ntegrated and they nfluence the relatve oston beteen the tool and the orkece. Effcently nvestgaton method of moton errors s not conformed enough so far. y exerment, t s sure that errors are varyng as the ork table rotates. The obectve of ths aer s ho to evaluate the oston-deendent geometrc errors of 5-axs machne tool and make error ma th the use of the measurement devce, called R-test.[ Keyords: R-test, 5-axs machne tool, Measurement, Poston-deendent geometrc errors.. Introducton A 5-axs machne tool has three lnear axes to translate the tool and/or the orkece and addtonal to rotary axes to tlt or rotate t. Due to ts caablty to machne more comlex orkeces, t has been oularzed radly n the manufacturng feld. Snce a 5-axs machnng center has lnear and rotary axes that are stacked over each other, moton errors of each axs and ts assembly error are accumulated n the ostonng error of a tool relatve to a orkece. onsequently, many machne tool users recognze that the machnng accuracy th 5-axs machne tool s loer than conventonal 3-axs machne tool. For mrovng the moton accuracy of 5-axs machne tool, t s frst crucal to develo a methodology to dentfy causes of the machne s moton error n an accurate and effcent manner. Statc oston and orentaton errors of the axs average lne of rotary axes are among the most fundamental error factors of the 5-axs knematcs, and called locaton errors n ISO 23-7[] or geometrc errors[2]. Many researches have been reorted on the dentfcaton of geometrc errors. Tycal one s the measurement methodology th the use of the double ball bar (D [3]. Ths methodology s currently under the dscusson at ISO T39/S2[4]. In the measurement th D devce, one ball s attached to the sndle and the other one s on the ork table, and the dstance beteen them s measured th a lnear encoder n the bar connectng to balls. Snce only one drecton dslacement s acqured n ths measurement, the oerator has to change the setu of the measurement devce and machne tool at least a coule of tmes to dentfy all the geometrc errors. Further more, t s dffcult for the measurement th the ball bar to measure more comlex error factors than geometrc errors. For examle, the ostonng error of rotary axes, the runout of rotary axes, the nfluence of the gravty are not ncluded n geometrc errors, and cannot be dentfed by ball bar measurement resented n [3]. To erform fully-automated calbraton of a larger class of error motons of 5-axs machnes, ball bar measurements[3] have nherent crtcal ssue. In ths study, e utlze the measurement devce called R-test[5][6]. It s commercally avalable by IS Precson Engneerng, Fda. The standardzaton of R-test has been also dscussed for ISO R-test can measure the oston of the tool n relatve to the orkece n three dmensons, hch allos us to obtan more data by one measurement setu than n the ball bar measurement. The obectve of ths aer s to develo a methodology to dentfy oston-deendent geometrc errors, or error ma, of rotary axes n 5-axs machne tools by usng the R-tests. 2. onfguraton of 5-axs Machne Tool and Error Parameters 2. onfguraton of 5-axs Machne Tool The confguraton of 5-axs machne tool consdered n ths aer s shon n Fg.. Ths machne tool has to rotary axes, and -axs, to tlt the ork table. When the - and Z-axes are arallel to each other as shon n Fg., the -axs s defned as Fgure : onfguraton of 5-axs machne tool 2.2 Poston-deendent Geometrc Errors As shon n Table, sx oston and orentaton error

2 Proceedngs of 4th IRP Internatonal onference on Hgh Performance uttng, 2 arameters are defned on each rotary axs. These arameters for the axs average lne of each rotary axs are defned n ISO23-7. Practcally, these error arameters may vary as the rotary axs rotates. In ths aer, these are consdered as a functon of the angular oston of rotary axs, and referred to as oston-deendent geometrc errors. Poston-deendent geometrc errors can model more comlex error motons. The errors on -axs are defned as a functon of the angular oston of -axs tself. The errors on -axs are defned as a functon of the - and -axes (e.g. δx (, δx (,. The errors on lnear axes are gnored n ths study, assumng that they are suffcently small n comarson th errors on rotary axes. Table : Defnton of error arameters of rotary axes δx ( Lnear shft of -axs from Y-axs n X drecton δy ( Lnear shft of -axs from Y-axs n Y drecton δz ( Lnear shft of -axs from Y-axs n Z drecton α ( Squareness error of -axs to Z-axs β ( Orentaton error of -axs around Y-axs γ ( Squareness error of -axs to X-axs δx (, Lnear shft of -axs from -axs n X drecton δy (, Lnear shft of -axs from -axs n Y drecton δz (, Lnear shft of -axs from -axs n Z drecton α (, Squareness error of -axs to -axs β (, Orentaton error of -axs around -axs γ (, Angular error of -axs 2.3 Knematc Modelng of 5-axs Machne Tool The knematc model to comute the relatve oston of the tool to the orkece s the bass of modelng the nfluence of moton errors on rotary axes to R-test measurements. Ths subsecton resents the model brefly. Defne the machne coordnate system as the coordnate system fxed on the machne frame, and the orkece coordnate system as the coordnate system fxed on the ork table. Homogeneous transformaton matrx (HTM to reresent the translaton to X, Y, and Z-drecton for the dstance of X, Y, and Z, and rotaton about X, Y, and Z-axs for the angle of A,, and, are shon as follos: X D ( X cos A sn A x D a ( A sn A cos A D y ( Y Y D b cos ( sn sn cos cos sn D z ( Z sn cos D Z c ( ( In ths aer, the commanded tool center ont n the orkece and machne coordnate systems s r resectvely descrbed as and. Its actual oston nfluenced by the moton errors are descrbed as r and. The left-sde subscrt r and resectvely reresents a vector n the machne and orkece coordnate systems. The rght-sde subscrt * reresents the commanded oston. When the commanded tool center ont n the orkece coordnate system at the angular command (, s gven by * [ * x, * y, * z ] T, t can be transformed n the machne coordnate system by the HTM as follos: T T r r ~ T (2 ~ here r T s the HTM that can transform from the tool center ont n the orkece coordnate system nto the machne coordnate system th angular commands: r ~ T Db ( Dc ( (3 Note that the dslacement sensors are fxed on the rotary table and tool center ont s measured n the orkece coordnate system. The relatve oston of the tool th resect to the sensors s reresented as: r r ~ ( T T (4 here r T s the HTM th error arameters n Table : r r T T T (5 r T D ( δx x D ( α a D y ( δy D ( β b D ( δz z D ( γ c D ( T Dx ( δx Dy ( δy Dz ( δz (7 Da ( α Db ( β Dc ( γ Dc ( Denotng [ x, y, z ] T, Eq. (4 can be rertten as: * x Δ Δ ΔX x * y Δ ΔA ΔY y (8 Δ Δ Δ * A Z z z When each error arameter s suffcently small, e have: ΔX { δy ( + δy (, }sn + { δx ΔY { δy { δx ΔZ δx ΔA { β + { α Δ { β { α Δ α ( cos δz ( + δy ( cos δz ( sn + δz ( + β ( cos γ ( + β ( cos γ ( sn + γ (, }cos (, }sn (, }cos ( cos + γ b ( sn + δx ( sn + δx ( cos + δz ( sn + α ( sn + α (, (, }cos (, }sn (, (, }cos (, }sn (6 (9

3 Proceedngs of 4th IRP Internatonal onference on Hgh Performance uttng, 2 3. Measurement th R-test devce 3. Outlnes of R-test Devce The overve of R-test devce s shon n Fg. 2. Ths devce s comosed of the ball attached on the machne sndle and three dslacement sensors attached on the rotary table. Sensors are fxed to drect aroxmately to the center of the ball. y dslacements measured by three sensors, the relatve devaton of the ball from the reference oston can be comuted. all attached on the Sndle Devce s attached on the table. Set on the Sndle of the machne tool Fgure 2: Overve of R-test devce Dslacement sensor 3.2 albraton of the Devce To get ball oston from dslacements of three sensors, the drecton vectors of each sensor need to be dentfed n advance. Furthermore, snce the center shft of the ball from the sndle centerlne sgnfcantly nfluences measurement results, t must be calbrated n ror and comensated. Frst, gve a set of certan knon command values to lnear axes and move the ball. Assumng that the ostonng error of lnear axes s neglgbly small, the drecton vector of each sensor can be dentfed from a set of measured ball dslacements. Then, by ndexng the sndle at a set of gven command angular ostons, get the dslacements of the sensors at each rotaton angle and dentfy the center shft of the ball. 3.3 Measurement rocedure In the measurement th R-test devce, the dslacement of each sensor s acqured at varous angular ostons of rotary axes. In ths study, -axs s set at every 3 from 9 to 9, and -axs s set at every 3 from to 33, to cover the entre movable range of rotary axes. The number of measurement onts s 84. The lnear axes are commanded such that the ball follos the sensors on the rotary table. When readng out the values of the sensors, all of the axes are rested. To observe the orentaton of -axs of rotaton, at least to sets of R-test measurements must be done th dfferent sensor locatons. More detals ll be gve n Secton Identfcaton of Poston-deendent geometrc errors 4. Identfcaton of errors on -axs In ths subsecton, the algorthm for dentfyng the oston-deendent geometrc errors of -axs s resented. Frst, t s assumed that all oston-deendent geometrc errors of -axs are zero. Errors of -axs at (~7 are descrbed as the vector: ( [ δx (, δy (, δz (, α (, β (, γ ( ] T ( The oston of the ball n the orkece coordnate system at and s gven by: ( [ ] T, x(, y(, z (, ( The Jacoban matrx reresentng the relaton beteen ( and, s gven by: x (, L δx (, (, y L (2 ( δx ( z (, L δx ( When all comonents of ( are suffcently small, the follong lnear aroxmaton s holds:,, ( (3 ( As descrbed n the revous secton, n R-test measurement, only the relatve dslacement of the ball from ts orgnal oston can be measured. In ths aer, for the smlcty of notaton, s defned as the orgnal oston. In other ords,, s alays [ ] T at. For ths reason,,,, (4 ( ( ( ( A set of oston-deendent geometrc errors, ( can be obtaned by solvng the follong otmzaton roblem by usng the least square method: mn 2 (, (, (, ( ( (5 ( ( Note that geometrc errors at, (, must be frst dentfed and then substtuted n Eq. (5 for other s. Equaton (8 reresents the relatonsh beteen the commanded oston * and measured oston n orkece coordnate system. y dfferentatng artally th each error arameter, the Jacoban matrx n Eq. (2 s gven. 4.2 Identfcaton of errors on -axs Defne the coordnate system that s fxed on the -axs and rotates concurrently th -axs rotaton as -axs coordnate system. In ths coordnate system, the commanded oston of the ball s gven by * (, (Eq. (6 and ts measured oston s gven by, (Eq. (7. (, (, Dc ( (6, Dc (, ( (7

4 Proceedngs of 4th IRP Internatonal onference on Hgh Performance uttng, 2 Identfy the oston-deendent geometrc errors on -axs by the follong rocedure. Frst, comute the nfluence of the oston-deendent geometrc errors of -axs dentfed n the revous subsecton n the -axs coordnate system as: ˆ, (, T (8 here T W s the HTM to transform the ball oston n the orkece coordnate system nto the -axs coordnate system gven by: T Db ( Dx( δx ( LDc ( γ ( Db ( Dc ( (9 Subtract ˆ (, from the measured oston by: q( (2, (, +, ˆ(, Ths reresents the ball oston that s nfluenced by only -axs errors. Ths rocess s conducted tce th the dslacement sensor located at dfferent ostons on the rotary table. As shon n Fg. 3, the orentaton of the lne connectng q (, and q 2 (, th resect to ther command ostons reresents the orentaton error of -axs. y roectng ths to Y-Z lane, α (, can be acqured. In the same ay, by roectng ths to X-Z lane, β (, can be acqured. In ths study, e assume that the angular ostonng error of -axs s suffcently small,.e. γ (,. Then, dentfy the translatonal errors on -axs. y usng acqured rotatonal errors of -axs, the nfluence of these errors to the ball oston n -axs coordnate system s smulated as follong: q (, (, Da ( α(, Db ( β(, Dc ( γ(, (2 Therefore, translatonal errors of -axs are gven by: δx (, δy (, q(, q (, (22 z(, δ α (, As descrbed n Secton 3.3, a set of R-test measurements as erformed on the exermental machne of the confguraton shon n Fg.. Then, comute the ball ostons by acqured sensor dslacements. Snce t s assumed that the moton errors on lnear axes are suffcently small n ths study, the ball dslacement measured by R-test,,, can be seen as the table dslacement by takng,. Table dslacements measured by the R-test rocedure are shon n Fg. 5. The ostons of sensor roected on X-Z lane at are shon n (a, and roected on the table surface at are shon n (b. In the same ay, the ostons of the sensor at 9 are shon n (c and (d. ommanded ostons of the sensor are reresented by, measured oston s reresented by. The devaton beteen commanded and measured ostons s magnfed by a factor of,. Fgure 4: ommands of each axs 2(, 2 q (, μm μm (, (a,on X-Z lane (b,on table surface q (, Z Fgure 3: omutaton of orentaton errors of -axs, for examle α (,, by means of to measurements. The rght-sde subscrt donets the frst measurement, 2 donates second measurement. The measured oston of the ball s reresented by, the smulated oston s reresented by. 5. Exermental ase Study 5. Measurement ommanded ostons of each axs are shon n Fg. 4. Y μm μm (c 9,on X-Z lane (d 9,on table surface Fgure 5: ommanded and measured table ostons by R-test measurement (Errors are magnfed by a factor of,.

5 Proceedngs of 4th IRP Internatonal onference on Hgh Performance uttng, Identfcaton of Poston-deendent Geomerc Errors and Smulaton Poston-deendent geometrc errors are dentfed from acqured oston errors by usng the algorthm resented n Secton 4. In Fg. 6, dentfed rotaton errors of -axs are reresented n (a, translatonal errors of -axs are n (b. Poston-deendent geometrc errors of -axs at and 9 are reresented n Fg. 7. To llustrate the nfluence of - and -axes error motons, the nfluence of dentfed oston-deendent geometrc errors of -axs only, shon n Fg. 6, s comuted by Eq. (8 and: r r T (23 here r T s gven by Eq. (5. The result s shon n Fg. 8. ommanded ostons of the sensor are reresented by, measured ostons are reresented by, and smulated ostons are reresented by. (a (a Rotatonal Errors (b 9 (b Translatonal Errors Fgure 6: Identfed oston-deendent geometrc errors on -axs 5.3 Observaton The follong observatons can be made: Identfed δx ( and δy ( (Fg. 6(b have an error about 4 μm and 6 μm, resectvely, n average. Ths s mostly caused by the ms-calbraton of -axs centerlne n X and Y drectons. The table oston at 9 s shfted by about μm to Z drecton. Fgure 7: Identfed oston-deendent geometrc errors on -axs Identfed δz ( s about 3 μm at, and get about 9 μm at 9. δz ( at ndcates the ms-calbraton of -axs centerlne n Z-drecton. If t s merely the ms-calbraton, δz ( must be the same for any angular ostons. The ncrease n δz ( at 9 suggests that the table s shfted n Z drecton, lkely due to the gravty nfluence. Identfed α ( (Fg. 6(a vares from 3-3 at 9 to +3-3 at 9. Identfed γ ( also

6 Proceedngs of 4th IRP Internatonal onference on Hgh Performance uttng, 2 shos analogous varaton. Ths suggests that the -axs has a conng error moton. Almost no angular oston error of -axs s observed (Fg. 6(a, β (. α (, and γ (, (Fg. 7 sho a snusodal error motons. Ths suggests that the -axs also shos a conng error moton. The amltude s smlar at and 9. μm μm (a, on X-Z lane (b, on table surface (c 9, on X-Z lane (d 9, on table surface Fgure 8: all ostons commanded, measured by R-test measurement, and smulated (Errors are magnfed by a factor of, 6. oncludng Remark The oston-deendent geometrc errors are dentfed effcently and automatcally th the use of R-test devce. When the error ma can be made th dentfed ostondeendent geometrc errors, the moton accuracy can be mroved by comensatng them. References [] ISO 23-7:26, Test code for machne tools Part 7:Geometrc accuracy of axes of rotaton. [2] Inasak, I., Kshnam, K., Sakamoto, S., Sugmura, N., Takeuch, Y., Tanaka, F., 997, Shaer generaton theory of machne tools ts bass and alcatons, Yokendo, Tokyo [n Jaanese]. [3] Kakno, Y., Ihara, Y., Shnohara, A., 99, Accuracy evaluaton method of N machne tool by D tests, REALIZE IN., [n Jaanese] [4] Tsutsum, M., Ihara, Y., Sato, A,. Mshma, N., Ibarak, S., Yamamoto, M., Kobayash, M., Yonetan, T., 28, Standardzaton of testng methods for knematc moton of fve-axs machnng centers Draft roosal for ISO standard --, Proc. of the 7th Manufacturng and Machne Tool onference, μm μm [5] rngmann,., Kna, W., 26, Model-based 'hase-the-all' albraton of a 5-Axes Machnng enter,annals of the IRP, 55/. [6] Wekert, S., 24, R-Test, a Ne Devce for Accuracy Measurements on Fve Axs Machne Tools, Annals of the IRP, vol.53/. [7] ISO 79- to 3: Test condtons for machnng centers, Part to 3.