Advanced Materals Research Vol. 874 (2014) 57-62 Onlne avalable snce 2014/Jan/08 at www.scentfc.net (2014) rans ech Publcatons, Swtzerland do:10.4028/www.scentfc.net/amr.874.57 Contour Error of the 3-DoF Hydraulc ranslatonal Parallel Manulator Ryszard Dndorf 1,a, Potr Wos 2,b 1 Kelce Unversty of echnology, al. ysacleca Panstwa Polskego, 25-314 Kelce, Poland 2 Kelce Unversty of echnology, al. ysacleca Panstwa Polskego, 25-314 Kelce, Poland a dndorf@tu.kelce.l, b wos@tu.kelce.l Keywords: mechatronc devces, translatonal arallel manulator, hydraulc manulator, electrohydraulc lnear axs, contour error. Abstract. he aer deals wth a selected roblem of contour error for the desred trackng trajectory of the satal 3-DoF hydraulc translatonal arallel manulator (PM). he rototye hydraulc PM conssts of a fxed base and a movng latform, that are connected by the jonts wth three hydraulc lnear axes. he closed-loo knematc chans of the hydraulc PM create structure 3-RRPRR, n that revolute jonts R and rsmatc jonts P ste out. he three equal ntegrated electro-hydraulc axes are rsmatc jonts P, n that hydraulc cylnders are ntegrated wth oston measurng systems and roortonal drectonal control valves. Introducton Parallel manulator are comuter-controlled mechatronc devces, whch have been used to assst humans n varous tasks [1,2]. In Deartment of Mechatronc Devces at Faculty of Mechatroncs and Machne Devces of Kelce Unversty of echnology (Poland) was constructed the three axs (3-axs) and three-degrees-of-freedom (3-DoF) hydraulc translatonal arallel manulator (PM), that conssts of a fxed base, movng latform and three ntegrated electrohydraulc axes, connected by the jonts. Overvew and 3D model of the rototye hydraulc PM was shown n Fgure 1. Fg. 1. Overvew (a) and 3D model (b) of the rototye hydraulc PM: 1 fxed base, 2,3,4 rotary mounts of hydraulc cylnders, 5,6,7 ntegrated electro-hydraulc axes, 8.9.10 jonts of movng latform, 11 n jonts. hree equal ntegrated electro-hydraulc axs create rsmatc jonts P of the hydraulc PM [3]. A sngle electro-hydraulc axs conssts of CS tye cylnder nternally ntegrated wth the magnetostrctve lnear oston sensors (Novostrctve ) and externally ntegrated wth 4/3-way hgh resonse drectonal valve drectly actuated wth electrcal oston feedback of tye 4WRSE. Hydraulc cylnder arameter: ston dameter D = 40 [mm], rod dameter d = 28 [mm], stroke All rghts reserved. No art of contents of ths aer may be reroduced or transmtted n any form or by any means wthout the wrtten ermsson of P, www.tt.net. (ID: 46.76.76.201-20/02/14,09:23:09)
58 erotechnology h = 220 [mm], nomnal ressure =16 [MPa]. he hydraulc cylnders was laced on the fxed base about crcumradus R = 250 [mm], the rod cylnders are connected by the jonts wth the movng latform about crcumradus r =130 [mm]. he maxmum length of the hydraulc axs carres out L max = L mn + h, where L mn = 425 [mm] s a ntal length. he closed-loo knematc chans of the hydraulc PM create structure 3-RRPRR, n whch revolute jonts R and rsmatc jonts P ste out. he 3D knematc model and knematc structure of the 3-RRPRR hydraulc PM was shown n Fgure 2. Fg. 2. he 3D knematc model (a) and knematc structure (b) of the 3-RRPRR hydraulc PM: R revolute jonts, P rsmatc jonts, CP tool center ont. Knematc of the hydraulc arallel manulator he comutatonal knematc model of 3-axs hydraulc arallel manulator shown n Fg.3. For the contour error of the 3-axs hydraulc arallel manulator the 6-DoF knematc model have been consdered. It ncludes both the latform oston varables x, y, z and also rotatonal RPY angles,,. he RPY (Roll-Ptch-Yaw) angles defned wth resect to three successve rotatons about the fxed X, Y, Z axes [4]. he roblem of nverse knematcs s to fnd the hydraulc cylnders elongaton L, gven the oston (x, y, z ) and orentaton error (,, ) of the CP. o solve the nverse knematcs roblem vectors r and R and matrx R A B are gven, then vector L from ont A A to ont B A can be calculated as: A B L R r R (1) where: = vector coordnates from ont B n movng frame to ont A n reference frame, [ x y z ], R A B rotaton matrx from the movng frame to the reference frame for gven values of RPY angles, r vector coordnates of B B n the movng frame (B, X, Y, Z): r c r s 0 r radus of crcumradus of the movng latform, r, R vector coordnates of A A n the reference frame (A, X 0, Y 0, Z 0 ): R c R s 0 R radus of crcumradus of the fxed late. R,
Advanced Materals Research Vol. 874 59 As shown n Fg.3, all connecton onts A of the bottom fxed late and connecton onts A of the to movng latform are fnd n the secfc locatons, as follows: for = 1,2,3, then 1 = 0 o,,. Fg. 3. Comutaton knematc model of 3-axs hydraulc arallel manulator. he length of vector L can be obtaned as: L L L (2) he arallel mechansm has the characterstc of closed loo, so the constrants of the mechansm movement could be exressed as f(l,q)=0. hen the nverse knematc soluton of a arallel manulator could be exressed as: dl J dq (3) where: Jacoban matrx J s obtaned as J 1 J L J q, L s a vector whch ndcates the varaton of hydraulc lnear axes, L L1 L2 L3, q s a vector whch ndcates the oston and rotaton of the movng latform, q x y z. he unque soluton of forward knematc of a oston and orentaton error of the movng latform are exressed as: q J 1 L (4) where: q [ x y z ], L L L ]. [ 1 2 L3
60 erotechnology he end-effector oston (x, y, z ) and orentaton error (,, ) of the movng latform could be obtaned by substtutng the elongaton L of every lnear hydraulc axs nto the forward moton equaton (Eq.(4). he Newton-Rahson (NR) method as an effectve numercal method can be aled to calculate the forward roblem of the hydraulc PM [4]. Contour error of hydraulc PM he exermental setu of the 3-axs hydraulc PM was shown n Fgure 4. he electro-hydraulc servo system s a strongly nonlnear object wth arameters changng over tme. he oston control system of the arallel manulator was aled to carry out the on-lne synthess of the roortonal ntegral dervatve (PID) controller. he mlementng the algorthm wth numercal smulaton and dentfcaton of the control arallel manulator were calculated [5,6]. he flow chart of closed-loo oston control system usng the trajectory lannng of the 3-axs hydraulc PM was shown n Fgure 4. Fg. 4. Exermental setu of the 3-axs hydraulc PM. Poston control system usng the trajectory lannng of the 3-axs hydraulc PM was shown n Fgure 5. Fg. 5. Poston control system usng the trajectory lannng of the 3-axs hydraulc PM. he desred sral trajectory of the movng latform (end-effector) used n exerments was generated: x =d sn(n/d); y =d cos(n/d)-10; z =L 0 +v z N for k=1, d=10, h=20, N=0:0.1:k 2d; v z =h/(k 2d); L 0 = 520 mm. he sral trajectory of the end-effector and the elongaton set of the hydraulc axs was shown n Fgure 6.
Advanced Materals Research Vol. 874 61 Fg. 6. Sral trajectory of the end-effector (a) and elongaton set of the hydraulc axs (b). Contour error of the end-effector oston x, y, z n coordnate XYZ wth the orentaton error for angle,, around the axs X, Y, Z take nto account was consder. In ths case the coordnate vector q has the form: q [ x y z ]. he orentaton error of the movng latform are comensated by control of the hydraulc cylnders. Contour error for sral trajectory between desred trajectory and actual trajectory on the lan X-Y was llustrated n Fgure 7a. he actual elongaton of the hydraulc axs after orentaton error comensaton was shown n Fgure 7b. Fg. 7. he contour error: a) sral trajectory on the lan X-Y: sold lne shows the desred ath, dashed lne shows the actual ath, b) actual elongaton of the hydraulc axs. Contour error s defned as the mnmum dstance from current actual oston to the desred contour [7,8]. he vector contour error e whch s from actual oston a to desred oston d was defne: e (6) a d he contour error vector s the shortest dstance between desred trajectory and actual oston. he contour errors n X, Y drecton are less than 0.5mm and 0.8mm and t can meet the requrements of the ractcal alcatons.
62 erotechnology Summary he contour trackng error of a 3-DOF hydraulc translatonal arallel manulator (PM) was nvestgated. In ths study, the ostons of movng latform are changed by extenson or shortenng of the three ntegrated electro-hydraulc axes. he errors n the outut ose of a manulator caused by dmensonal errors, such as these n hydraulc cylnders and lnks, can be classfed nto two grous: comensatable errors and uncomensatable errors. hey deend on knematc structures and arameters of the hydraulc PM. For examle, f there s a dmensonal error n a lnk of a connectng chan of a PM, the latform changes ts orentaton accordng to moton. hs orentaton error cannot be comensated for by the nut of actve jonts (hydraulc cylnders). o ncrease robot accuracy, the above descrbed errors should be comensated. he straghtforward aroach n robot error comensaton s based on the absolute oston feedback, that s rovded by addtonal sensors that are able to measure the oston and orentaton of the CP wth resect to about the robot base coordnate frame. References [1] L.-W. sa, Robot Analyss: he Mechancs of Seral and Parallel Manulators, John Wley & Sons, New York, 1999. [2] J.-P. Merlet, Parallel Robots, Sold Mechancs and ts Alcatons, Srnger Verlag, New York, 2006. [3] R. Dndorf, P. Wos, Integrated electro-hydraulc systems, n: Monograh Cylnder Lbrary, Komag, Glwce, 2011,.101-107 (n Polsh). [4] C.C. Nguyen, S. Antraz, Z.L. Zhou, C. Cambell, Adatve Control of a Stewart Platform- Based Manulator, J. Robotc Syst. 10 (1993) 657-687. [5] P. Wos, R. Dndorf, Adatve control of the electro-hydraulc servo-system wth external dsturbances, Asan J. Control 15 (2013) 1065-1080. [6] P. Woś, R. Dndorf, Adatve control of a arallel manulator drven by electro-hydraulc cylnders, I.J.A.M.E. 17 (2012) 1061-1071. [7] Y. J. Lou, Z. B. L, Y. G. Zhong, J. G. L, Z. X. L, Dynamcs and contourng control of 3-DOF arallel knematcs machne, Mechatroncs 21 (2011) 215-226. [8] A. Klmchk,A. Pashkevch, D. Chablat, G. Hovland, Comlance error comensaton technque for arallel robots comosed of non-erfect seral chans, J. Robot. Comut. Integr. Manuf. 29 (2013) 385-393.
erotechnology 10.4028/www.scentfc.net/AMR.874 Contour Error of the 3-DoF Hydraulc ranslatonal Parallel Manulator 10.4028/www.scentfc.net/AMR.874.57