ERROR MODELING FOR STRUCTURAL DEFORMATIONS OF MULTI-AXIS SYSTEM BASED ON SVR

Journa of Theoretca and Apped Informaton Technoogy 3 st January 03. Vo. 47 No.3 005-03 JATIT & LLS. A rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 ERROR MODELING FOR STRUCTURAL DEFORMATIONS OF MULTI-AXIS SYSTEM BASED ON SVR ZHENZHONG LIU, LIANYU ZHAO, ZHI BIAN Schoo of Mechanca Engneer, Tanjn Unversty of Technoogy, Tanjn 300384, Chna ABSTRACT Mut-axs system s consttuted by the mechanca nks, and the nk poston - oad deformatons are nonnear. That s an mportant factor to affect the postonng accuracy of mut-axs system. As the nonnear characterstcs of nk poston and oad deformatons, a nove method for modeng deformatons of oaded components based on SVR was presented. The R manpuator oad deformatons error modes were budng by ε-support vector regresson, and accordng to the mode, the trajectory error compensaton was smuated. The smuaton of structure deformatons modes and moton error compensaton of R manpuator shows the SVR method coud be used to bud the deformatons error mode effectvey. Keywords: Mut-axs System, Structura Deformaton, SVR. INTRODUCTION In manufacturng systems, the structura deformatons caused by the stffness are an mportant factor to affect the system postonng error. Mut-axs system s consttuted by the mechanca nkage, and the nk poston - oad deformatons are non-near. The nk poston - oad deformatons mode woud be dffcut to be derved by smpe measurement or mathematca methods. The commony used methods of structura deformatons modeng are the drect method [], the stffness matrx method [-4], the fnte eement method [5-7], the expermenta method [8] and the synthess method [9]. L Bng [] estabshed the man stffness mode of varabe axs CNC usng the drect method. The method s smpe, but the versatty s mted, ony appcabe to the stffness modeng of the standard poston and orentaton. Based on nfuence coeffcent method and prncpe of vrtua work, Zhao Tesh [] estabshed contnuous stffness nonnear mappng genera mode of spata parae mechansm, ncudng the eastc deformatons of actve and passve hnge and gravtes factor. Combnng the stffness matrx, the performance ndex k used to estmate the mechansm stffness s defned, but t cannot come to the exact souton. Deng Yaohua [5] studed predcton of fexbe matera deformatons usng spne fnte eement method. Ths method has a good cacuaton speed, but arge error s exsted wth the practca engneerng. Gossen [8] through the statc stffness experments soved the machne statc stffness by the method of oadng and measurement deformatons. It s a way to objectvey refect the actua workng condtons, but the expermenta method s a arger workoad. Cnton [9] assumed near reatonshp between the stffness and ength. Wth the mnmum average error as the optmzaton goa, the stffness cacuaton mode was obtaned by 4 measurement data, but n fact the stffness and ength s often non-near. For the nonnear reatonshp between the nk poston and ts defecton of the nonnear transmsson mechansm of the mut-axs system, ths paper proposed the oad deformatons error modeng method based on support vector regresson [0], and estabshed the nk poston and oad deformatons mode wth support vector regresson agorthm.. CHARACTERISTICS OF POSITION AND LOAD DEFORMATIONS Fgure shows the stretchng and bendng of two dfferent force condtons of the nk. Fgure (a) s a stretchng rod. The ength s L. The constant crosssectona area s A. The weght sw, and the force s P. Fgure (b) s a cantever beam, bendng force P and unform oad of q = W / L. The sze of the beam and fgure (a) are the same. 058

Journa of Theoretca and Apped Informaton Technoogy 3 st January 03. Vo. 47 No.3 005-03 JATIT & LLS. A rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 E,A,I W L q=w/l E,A,I P The sampe data ( x, y),,( x, y) ( X R) s known. And ε-nsenstve error oss functon metrcs of observed vaue y and functon predctve vaue of f( x) = w φ( x) + b, that s: 0 y f( x, x) = max ε y f( x, x) ε (3) P (a) Fgure : Force Condtons Of Lnks By the matera mechancs, the ongtudna deformatons of fgure (a) under the acton of the tenson P and gravty W are as foows: In the formua, L (b) a a Y = PL / ΕΑ, Y = WL /ΕΑ () p q a Y p represents a P generated a under tense deformatons; Y q s the generated tense deformatons under the acton of the unform oad W ; E s the moduus of eastcty of the matera. The bendng deformatons of the nk: Y = PL /3 ΕΙ, Y = WL /8ΕΙ () b 3 b 3 p q b Wheren, Y p s the bendng deformatons under b the acton of the force P ; Y q s the bendng deformatons under the acton of the unform oad q ; and I denotes the moment of nerta. By the formua () and (), the nk poston and the nk deformatons are non-near when the change n the ange of the nk wth respect to the horzonta pane. I.e., the deformatons of these components and the oad on whch are not drecty proportona. The ssue of the machnng and assemby of parts cannot be guaranteed the nk oad s n the center ne, the nk but aso by the nfuence of the torsona oad, n most cases. Therefore, the deformatons of the nk rod are not easy obtaned by a drect cacuaton method, and by actua measurement to obtan the accurate amount of deformatons. 3. THE SUPPORT VECTOR REGRESSION MODEL ALGORITHM In ths paper, ε-support vector regresson buds the modes of the deformatons of the nk rgdty, and ts prncpes are as foows: The sack varabe ξ () = ( ξ, ξ,, ξ, ) T ξ and the penaty functon C are ntroduced. Then, the support vector machne regresson transforms nto mathematca optmzaton probem: mn ( w w) + C ( ) ξ + ξ w, ε, ξ, b ( w φ( x) + b) y ε + ξ, st.. y ( w φ( x) + b) ε + ξ, ξ, ξ 0 Introducng Lagrange functon, Lwb w w C () () () (,, ξ, α, η ) = ( ) + ( ξ + ξ ) (4) ( ηξ ηξ ) α( ε ξ y w φ( x) b) + + + α ( ε + ξ y + w φ( x ) + b) Wheren, () (5) α = ( α, α,, α, α ) T and () η = ( η, η,, η, ) T η are Lagrange mutper vectors. Accordng to Fermat theorem, L = w = w L = ( α α ) = 0 b L () () = C η α = 0 ξ α () () () ( α α)( φ x) 0 0, η 0 (6) Then, the dua form of optmzaton probem (4) s mn [ α ( y ε) α( y + ε)] αα, ( α α )( α j α ) K( x, xj) j= st.. ( α α ) = 0, 0 α, α C/,,, (7) 059

Journa of Theoretca and Apped Informaton Technoogy 3 st January 03. Vo. 47 No.3 005-03 JATIT & LLS. A rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 In the formua, K( x, x ) = φ( x ) φ( x ) s the j j kerne functon. The orgna probem s transformed nto the convex quadratc programmng of the equaton (7), and the souton s ( αα, ), thus the regresson equaton: f x = w x + b = K x x + b (8) ( ) φ( ) ( α α ) (, j) SV Wheren, ( α α) (, j) εα, j (0, / ) k α α k εα j b = y K x x + C b = y ( ) K( x, x ), (0, C/ ) 4. THE ESTABLISHMENT OF THE TRAINING SAMPLES (9) Ths paper descrbes the stffness deformatons modeng method based on SVR, wth R manpuator as the research object shown n Fgure. Fgure : The Force Condtons Of The R Manpuator Lnk The DH parameters of the two nks of the R manpuator are consstent. The nks are hoow ppes. The ength of the two nks s 600mm. The nner crce of the two nks s 60mm. The OD of the two nks s 80mm, and the matera of the two nks s 45 # stee. The oad deformatons are anayzed by the fnte eement software Ansys to obtan the sampe data. Make nk and nk of R manpuator respectvey wth the Cartesan coordnates X axs from 0 to 90. The end (nk) of R manpuator s apped from 0N to 60N wth oad P. The oad drecton of the nks s perpendcuar to the X axs. Consderng nfuence of ts own gravty, the oad range of the nk at the jonts B s from 0N to 6N. The oad deformatons of nk and nk are anayzed, as shown n tabe and tabe. m θ P A Z X m B θ X Tabe : Deformatons Anayss Resuts Of Lnk (Unt: Mm) 0N N N 3N 4N 5N 6N 0 0.03697 0.03963 0.0456 0.044899 0.04754 0.05085 0.0588 0 0.03644 0.03907 0.046 0.0444 0.04687 0.04943 0.05033 0 0.03475 0.03736 0.0397 0.0405 0.044689 0.04774 0.049658 30 0.03033 0.03433 0.03663 0.038904 0.0494 0.043484 0.045774 40 0.0834 0.030368 0.03394 0.0344 0.036447 0.038473 0.040499 50 0.03789 0.0549 0.079 0.0889 0.03059 0.0393 0.033994 60 0.0854 0.09837 0.06 0.0485 0.03809 0.053 0.06456 70 0.078 0.03694 0.04608 0.055 0.06435 0.07349 0.0863 80 0.006507 0.006973 0.007438 0.007904 0.008369 0.008834 0.0093 90 0.000 0.0006 0.0004 0.00054 0.00068 0.0008 0.00097 060

Journa of Theoretca and Apped Informaton Technoogy 3 st January 03. Vo. 47 No.3 005-03 JATIT & LLS. A rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 Tabe : Deformatons Anayss Resuts Of Lnk (Unt: Mm) 0 N 0 N 0 N 30 N 40 N 50 N 60 N 0 0.00967 0.058 0.04699 0.074 0.0978 0.03 0.04863 0 0.00947 0.0974 0.04477 0.0698 0.09483 0.0985 0.04489 0 0.009039 0.047 0.0386 0.0604 0.08593 0.0098 0.0337 30 0.00833 0.00533 0.0734 0.04936 0.0737 0.09338 0.054 40 0.00737 0.00938 0.066 0.033 0.056 0.0708 0.09057 50 0.00686 0.0078 0.009455 0.009 0.075 0.04358 0.05994 60 0.00483 0.006085 0.007357 0.00869 0.00990 0.073 0.0445 70 0.00395 0.00466 0.005036 0.005907 0.006777 0.007648 0.00859 80 0.00677 0.00 0.0056 0.003005 0.003448 0.00389 0.004334 90 6.5E-05 7.8E-05 9.3E-05.04E-04.7E-04.3E-04.44E-04 5. TRAIN AND TEST OF THE STRUCTURE DEFORMATIONS MODELS BASED ON SVR Based on support vector regresson agorthm, the sampes were traned by the bsvm toobox []. The kerne functon K( x, x j) was eected the RBF kerne functon: x x K( x, x) = exp (0) σ By experence method, the punshment factor C s 50, and the RBF kerne functon wdth σ s 4. The tabe and tabe are the earnng sampes of error mode, and the earnng sampes are as the test sampe. The fttng resuts are as shown n tabe 3 and tabe 4. Tabe 3: The Fttng Vaues Of Lnk Based On SVR (Unt: Mm) 0N N N 3N 4N 5N 6N 0 0.03697 0.03963 0.0456 0.044899 0.04754 0.05085 0.0588 0 0.03644 0.03907 0.046 0.0444 0.04687 0.04943 0.05033 0 0.03475 0.03736 0.0397 0.0405 0.044689 0.04774 0.049658 30 0.03033 0.03433 0.03663 0.038904 0.0494 0.043484 0.045774 40 0.0834 0.030368 0.03394 0.0344 0.036447 0.038473 0.040499 50 0.03789 0.0549 0.079 0.0889 0.03059 0.0393 0.033994 60 0.0854 0.09837 0.06 0.0485 0.03809 0.053 0.06456 70 0.078 0.03694 0.04608 0.055 0.06435 0.07349 0.0863 80 0.006507 0.006973 0.007438 0.007904 0.008369 0.008834 0.0093 90 0.000 0.0006 0.0004 0.00054 0.00068 0.0008 0.00097 06

Journa of Theoretca and Apped Informaton Technoogy 3 st January 03. Vo. 47 No.3 005-03 JATIT & LLS. A rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 Tabe 4: The Fttng Vaues Of Lnk Based On SVR (Unt: Mm) 0 N 0 N 0 N 30 N 40 N 50 N 60 N 0 0.00967 0.058 0.04699 0.074 0.0978 0.03 0.04863 0 0.00947 0.0974 0.04477 0.0698 0.09483 0.0985 0.04489 0 0.009039 0.047 0.0386 0.0604 0.08593 0.0098 0.0337 30 0.00833 0.00533 0.0734 0.04936 0.0737 0.09338 0.054 40 0.00737 0.00938 0.066 0.033 0.056 0.0708 0.09057 50 0.00686 0.0078 0.009455 0.009 0.075 0.04358 0.05994 60 0.00483 0.006085 0.007357 0.00869 0.00990 0.073 0.0445 70 0.00395 0.00466 0.005036 0.005907 0.006777 0.007648 0.00859 80 0.00677 0.00 0.0056 0.003005 0.003448 0.00389 0.004334 90 6.5E-05 7.8E-05 9.3E-05.04E-04.7E-04.3E-04.44E-04 Compare tabe and tabe wth tabe 3 and tabe 4, we can get: the errors between the fttng vaues by SVR and the Ansys smuaton vaues are very sma, whch can be negected. The SVR has exceent functon approxmaton abty. The fttng vaues of nk by the east square method are as shown n tabe 5. Tabe 5: The Fttng Vaues Of Lnk By The Least Square Method (Unt: Mm) 0 N 0 N 0 N 30 N 40 N 50 N 60 N 0 0.00960 0.038 0.04675 0.07 0.09750 0.086 0.0483 0 0.009490 0.0998 0.04505 0.0703 0.095 0.007 0.04535 0 0.009054 0.0446 0.03839 0.063 0.0863 0.004 0.03408 30 0.008330 0.0053 0.0733 0.04934 0.0735 0.09335 0.0538 40 0.007356 0.009300 0.044 0.0388 0.053 0.07075 0.0900 50 0.00669 0.007800 0.009430 0.006 0.069 0.043 0.0595 60 0.004806 0.006077 0.007347 0.00868 0.009889 0.059 0.0430 70 0.003307 0.0048 0.005054 0.00597 0.00680 0.007675 0.008549 80 0.00707 0.0057 0.00606 0.003056 0.003505 0.003955 0.004405 90 4.46E-05 5.9E-05 6.E-05 6.9E-05 7.73E-05 8.65E-05 9.48E-05 Contrast tabe 4 and tabe 5, we can obtan the errors are bgger by the east square method, and they are between 0 µm to 0 µm. 6. SIMULATION OF TEST SYSTEM 6. System Components As shown n Fgure, the R robot s the compensaton vadaton object of the deformatons error modes, and ts forward knematcs are: 06

Journa of Theoretca and Apped Informaton Technoogy 3 st January 03. Vo. 47 No.3 005-03 JATIT & LLS. A rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 X = cosθ + cos( θ + θ ) Z = snθ + sn( θ + θ ) The nvers knematcs are: θ = π + + arccos( ( X Z ) / ) () θ = arctan( Z / X) arctan( sn θ /( + cos θ)) () 6. Verfcaton of the Structura Deformatons Mode 6.. Expermenta scheme R robot trajectores run for some dstance aong the Z-axs drecton. The structura deformatons errors of each nk n the trajectory are anayzed based on SVR. Thus compensate for the error, and verfy the vadty of the structura deformaton error mode support vector regresson anayss. 6.. Expermenta procedure Let R manpuator trajectory run n the Cartesan coordnate, as [80, 0, 600] to [80, 0, 800] (unt: mm) of the straght ne segments. Wthout consderng the stffness deformatons, the anges of θ and θ are shown n Fgure 3 and Fgure 4. The anguar dspacement of nk ( ) The anguar dspacement of nk ( ) 30 5 0 5 0 5 0 600 60 640 660 680 700 70 740 760 780 800 The end Z-axs coordnate (mm) 65 60 55 50 45 40 35 Fgure 3: θ Ange Curve 30 600 60 640 660 680 700 70 740 760 780 800 The end Z-axs coordnate (mm) Deformatons of nk (mm) Deformatons of the end (mm) Deformatons of nk (mm) 0.06 0.05 0.04 0.03 0.0 0.0 +: Cacuated vaues of Ansys; O: Ftted vaues 0 0 0 0 30 40 50 60 70 80 90 Anguar dspacement of nk ( ) Fgure 5: Deformatons Curve Of Lnk 0.06 +: Cacuated vaues of Ansys; O: Ftted vaues 0.04 0.0 0.0 0.08 0.06 0.04 0.0 0.0 0.008 0.006 0.004 0.00 0 0 0 0 30 40 50 60 70 80 90 Anguar dspacement of nk ( ) Fgure 6: Deformatons curve of nk 0.07 0.065 0.06 0.055 0.05 0.045 600 60 640 660 680 700 70 740 The end Z-axs coordnate (mm) Fgure 7: Error-Dspacement Curves Before Compensaton Then the error-dspacement curve of R manpuator can be obtaned, as shown n Fgure 7. The stffness error mode based on SVR can be reazed trajectory error oop compensatng, and the agorthm fowchart s shown n Fgure 8. The error-dspacement curve after compensaton s shown n Fgure 9. Fgure 4: θ Ange Curve Assumng that the end oad of the R robot s 60N, based on the aforementoned SVR modeng method, we can get the error modes of nk and nk as shown n fgure 5 and fgure 6. The abscssa represents the ange of the nk wth the X axs, the ordnate s the amount of eastc deformatons. 063

Journa of Theoretca and Apped Informaton Technoogy 3 st January 03. Vo. 47 No.3 005-03 JATIT & LLS. A rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 Trajectory dscretzaton effectve used to bud the structure error compensaton for mut-axs system. Inverse knematcs Error modeng based on SVR ACKNOWLEDGEMENTS Ths work was supported by Tanjn Scence and Technoogy Deveopment Fund Project for Coeges and Unverstes under Grant No. 0040. ΔZ Δ Yes Increments of each jont R manpuator No Z = Z +ΔZ Fgure 8: The Agorthm Fowchart Of Error Loop Compensatng Deformatons of the end (mm) x 0-3 0.8 0.6 0.4 0. 0-0. -0.4-0.6-0.8-600 60 640 660 680 700 70 740 760 780 800 The end Z-axs coordnate (mm) Fgure 9: Error-Dspacement Curve After Compensaton Fgure 8 show that the structura deformatons have a arge mpact on mutaxa nonnear nk postonng accuracy. The structura deformatons must be compensated. Fgure 9 shows that the SVR can be effectve used to the structure error compensaton for mut-axs system. The structura deformaton of the cosed-oop error compensaton can be acheved. 7. CONCLUSION The structura deformatons have a arge mpact on mutaxa nonnear nk postonng accuracy, and the structura deformatons shoud be compensated. The SVR has exceent functon approxmaton abty. It s better than the east square method. The smuaton of structura deformatons modes and moton compensaton of R manpuator shows the SVR method coud be REFERENCES: [] L Bng, Wang Zhxng, Hu Yng, The stffness cacuaton mode of the new typed parae machne too, Journa of Machne Desgn, Vo. 3, No. 3, 993, pp. 4-6. [] Zhao Tesh, Zhao Yanzh, Ban Hu, et a, Contnuous stffness nonnear mappng of spata parae mechansm, Chnese Journa of Mechanca Engneerng, Vo. 44, No. 8, 008, pp. 0-5. [3] Wang Youyu, Huang Tan, Chetwynd D G, et a, Anaytca method for stffness modeng of the trcept robot, Chnese Journa of Mechanca Engneerng, Vo. 44, No. 8, 008, pp. 3-8. [4] Anatoy Pashkevch, Damen Chabat, Phppe Wenger, Stffness anayss of overconstraned parae manpuators, Journa of Mechansm and Machne Theory, Vo. 44, No. 5, 009, pp. 966-98. [5] Deng Yaohua, Lu Guxong, Wu Lmng, et a, Deformaton predcton of fexbe workpeces and an error compensaton method n NC machnng processng, Mechanca Scence and Technoogy for Aerospace Engneerng, Vo. 9, No. 7, 00, pp. 846-85. [6] Peng Zh, Wang Lpeng, Wang Xnyan, The FEM anayss for gudeway deformaton advance compensaton of CNC machne too, Machne Toos & Hydraucs, Vo. 39, No., 0, pp. 6-7. [7] Bashar S. E., PLACID M. F., Computaton of stffness and stffness bounds for parae nk manpuators, Internatona Journa of Machne Toos & Manufacture, Vo. 39, No., 999, pp. 3-34. [8] Cément M Gossen, Stffness mappng for parae manpuators, IEEE Transactons on Robotcs and Automaton, Vo. 6, No. 3, 990, pp. 377-38. 064

Journa of Theoretca and Apped Informaton Technoogy 3 st January 03. Vo. 47 No.3 005-03 JATIT & LLS. A rghts reserved. ISSN: 99-8645 www.jatt.org E-ISSN: 87-395 [9] Chares M Cnton, Guangmng Zhang, Stffness modng of a stewart-patform-based mng machne, Trans of the North Amerca Manufacturng Research Insttuton of SME, No. 5, 997, pp. 335-340. [0] Chapee O, Choosng mutpe parameter for support vector machnes, Machne Learnng, Vo. 46, No., 00, pp. 3-59. [] C-C Chang, C-J Ln, LIBSVM : a brary for support vector machnes, ACM Transactons on Integent Systems and Technoogy, 0, pp. -7. 065