Contour Error of the 3-DoF Hydraulic Translational Parallel Manipulator. Ryszard Dindorf 1,a, Piotr Wos 2,b

Size: px

Start display at page:

Download "Contour Error of the 3-DoF Hydraulic Translational Parallel Manipulator. Ryszard Dindorf 1,a, Piotr Wos 2,b"

Aubrie Jefferson
6 years ago
Views:

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.

1 Advanced Materals Research Vol. 874 (2014) Onlne avalable snce 2014/Jan/08 at (2014) rans ech Publcatons, Swtzerland do: / 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, Kelce, Poland 2 Kelce Unversty of echnology, al. ysacleca Panstwa Polskego, 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, 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, (ID: /02/14,09:23:09)

2 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 Advanced Materals Research Vol 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

4 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.

5 Advanced Materals Research Vol 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.

6 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, [2] J.-P. Merlet, Parallel Robots, Sold Mechancs and ts Alcatons, Srnger Verlag, New York, [3] R. Dndorf, P. Wos, Integrated electro-hydraulc systems, n: Monograh Cylnder Lbrary, Komag, Glwce, 2011, (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) [5] P. Wos, R. Dndorf, Adatve control of the electro-hydraulc servo-system wth external dsturbances, Asan J. Control 15 (2013) [6] P. Woś, R. Dndorf, Adatve control of a arallel manulator drven by electro-hydraulc cylnders, I.J.A.M.E. 17 (2012) [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) [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)

7 erotechnology / Contour Error of the 3-DoF Hydraulc ranslatonal Parallel Manulator /

ON THE DESIGN OF LARGE SCALE REDUNDANT PARALLEL MANIPULATOR. Wu huapeng, Heikki handroos and Juha kilkki

ON THE DESIGN OF LARGE SCALE REDUNDANT PARALLEL MANIPULATOR Wu huapeng, Hekk handroos and Juha klkk Machne Automaton Lab, Lappeenranta Unversty of Technology LPR-5385 Fnland huapeng@lut.f, handroos@lut.f,