Enhancing Parallel Robots Accuracy with Redundant Sensors

Transcription

1 Enhancng Parallel obots Accuracy wth edundant ensors Frédérc Marquet, Olver Company, ébasten Krut, Franços Perrot LIMM - UM 556 CN Unversté Montpeller rue Ada, Montpeller Cede 5, France <marquet, company, krut, perrot >@lrmm.fr Abstract. hs paper ntroduces a control strategy based on redundant sensors that leads to parallel robots accuracy enhancement. he method s presented n general, then appled to a 4-dof parallel robot. Practcal mplementaton ssues, smulaton results and epermental valdaton are addressed. 1. Introducton After the frst deas of parallel mechansms proposed by Gough [1 or tewart [2, ntroducng the dea that an ecellent stffness could be obtaned wth PKM (Parallel Knematc Machnes), Clavel proposed n the late 8 s the famous Delta structure [3 as a base for a famly of parallel machnes dedcated to hgh-speed applcatons. hus many PKM have been used for pck-and-place and more recently for machnng: n both applcaton domans, PKM are consdered as more accurate than ther seral counterparts; n fact, ths ssue s controversal: by prncple, PKM should be more accurate (because actuated jonts postonng errors do not add to each others [4), but n practce t s not always the case (because PKM nvolve many passve jonts leadng to a comple error model). Consequently, n recent researches, a lot of effort has been dedcated to PKM calbraton ([5[6), wth a very promsng trend to provde self calbraton technques ([7[9), and a contnuous fundamental work for refnng the mathematcal tools nvolved n calbraton ([8[1). uch works have dramatcally mproved the obtaned performances regardng PKM accuracy, but few drawbacks reman: any calbraton procedure s tedous, whatever the measurement systems are, and even the best calbraton results are not guaranteed for the complete machne lfe cycle. he track whch s proposed to consder here s dfferent: mprovng PKM accuracy by usng real-tme redundant measurements. In prevous works, redundant poston sensors have been mostly used as a convenent mean for solvng forward knematc problems, and the trend s to try to reduce as much as possble the number of redundant sensors needed to solve such problems ([11[12[13[14). However, for most parallel applcatons, solvng the Forward Knematc problem s only a small part of the control needs, and t s often run at the user request only (to get the knowledge of the robot actual poston) and not n real-tme. Indeed the typcal control scheme of a parallel robot s represented n Fgure 1. he trajectory s frstly generated n the Cartesan space; then each Cartesan locaton s transformed nto a poston vector n the jont space thanks to the Inverse Knematc Model (IKM); the control (that s, the hard real-tme part of the process) s fnally done n that latter space; the Forward Knematc Model (FKM) s often mplemented as a HMI routne, or only runs at ntalzaton phase. raj. d q d q Gen. IKM + C OBO - GUI, f... FKM Fgure 1. ypcal PKM control archtecture Of course, redundant sensors can help to solve the FKM by offerng closed form solutons by speedng up an teratve numercal computaton; ths pont wll be brefly recalled and addressed for our specfc case. However the pont of ths paper s to use one opportunty offered by PKM: they are bult wth many passve jonts that are often deal locatons to place sensors at low addtonal costs. On the opposte of the tradtonal pontof-vew, t s proposed here to use as many addtonal sensors as possble, and to nclude them n a real-tme control scheme consstent wth the usual soluton depcted n Fgure 1. In the followng sectons, the proposed strategy s frstly descrbed, then smulated, and ts mplementaton on a specfc 4-degree-of-freedom (dof) PKM s eplaned. mulaton and epermental results let us epect that sensor redundancy could be useful to mprove the accuracy of PKM n ther daly use. q 2. Use of redundant sensors data n the PKM control scheme Poston errors occur for many reasons on a PKM: Œ Control errors; bascally, the actuator s not at the rght poston at the rght tme; ths s measured by the usual poston sensors and s supposed to be managed by controllers;

2 Œ Œ raj. Gen. GUI Knematc model errors; for practcal reasons, actual szes and relatve locatons of all lnks and jonts are not eactly as envsoned when the PKM was desgned. hs s not measured by the usual poston sensors, and s often consdered as a calbraton problem, but t could be sensed by sensors placed on passve jonts. Hgher order errors; ths could come from mechancal or thermal deformatons, or materal agng leadng to ncreased backlash for eample. hs s etremely comple to model, and thus etremely comple to compensate by calbraton; agan, dependng on sensors resoluton, part of those errors could be sensed by redundant sensors. d, f... IKM KM FKM q d r d Fgure 2. Archtecture of the proposed control strategy Name ze Descrpton d n 1 Nacelle desred poston q d n 1 Motors desred poston r d r 1 Desred redundant postons q de ( n + r) 1 Desred etended postons n 1 Nacelle poston q e ( n + r) 1 Etended postons qe ( n + r) 1 Etended errors ˆ q n 1 Modfed errors n ( n + r Etended to jont transformaton matr M M + = ( d, d ) EJ ) q de + - able 1. Varables defnton ˆq EJ C OBO In the proposed strategy (Fgure 2), the Cartesan rajectory Generaton module (that s the desred Cartesan locaton) feed two knematc models: Œ he classcal Inverse Knematc Model gvng the set of actuated jont postons; for a n-motor robot, the IKM provdes a n1 vector; Œ A edundant Knematc Model (KM), whch gves the set of correspondng redundant postons; f r redundant measurements are avalable, the KM provdes a r1 vector (able 1). As stated above, the easest way to get redundant measurements s to q e q e nstall poston sensors on some of the robot passve jonts; ndeed, those redundant measurements can be done by eternal means as well (e.g.: drect measure of the PKM nacelle poston along one drecton wth a telemeter, ) Both poston vectors are then grouped to form a sngle (n+r)1 vector, called the vector of etended postons. hs vector s the controlled varable, and s compared to a correspondng vector of etended sensor measurement. Dfferent solutons could be used to go from the (n+r)1 error vector to the n1 command vector to be sent to the motors amplfers. It s proposed here to derve a matr mappng the etended error vector nto a vector of modfed actuated jont errors; then classcal controllers could be appled. Obvously, t s stll possble to compute the FKM when requred by the robot user. In the followng sectons, a practcal mplementaton of the above deas wll be presented. 3. Applcaton to the H4 robot H4 s a 4-dof PKM (Fgure 3) whose mechancal desgn belongs to the famly of Delta and Hea robots. Many passve jonts are used for such mechansms (U jonts, ball jonts, etc.) and dependng on the technology, t could be easy or not to add redundant sensors on those passve jonts. For eample, t s clear on the archtectural scheme n Fgure 4 that the three revolute passve jonts could be easy-and-low-cost canddates for a redundant sensor. One could also nstall a telemeter on the robot base to measure one dstance between base and nacelle: ths could gve nformaton about the poston along z as. It has been decded to place only one redundant sensor on the pvot jont on whch the tool s mounted. DD Motor wo sphercal jonts wo sphercal jonts wo passve jonts Fgure 3. CAD model of the frst H4 prototype

3 4 :1 edundant measure Fgure 4. Archtectural scheme of the H4 prototype At ths stage, the IKM, KM and JE relatons have to be derved. - IKM In ths secton the relaton between actuators and nacelle poston represented by, respectvely, q = [ q 1, q2, q3, q4 and = [, y, z, θ s derved. 2d P 2 P 3 α 3 motors locatons α 2 2h u z u y u α 4 u 1 A 2 C 2 A 3 α 1 h P 1 P 4 D Fgure 5. Desgn parameters he selected desgn s descrbed n Fgure 5, where the followng parameters have been chosen: α 1 =, α2 = π, α3 = 3π / 2 α4 = 3π / 2 d u z O P u y u A 1 C 1 A 4 θ A L motor u l nacelle angle arm q B forearm And f w s the vector from P to A, t has been demonstrated that ths equaton leads to: M cos q + N sn q = G (1) where: M = 2l (( w ) cosα + ( w u y ) snα ) N = 2l ( w ) G = L² l² w ² esortng to the followng new varable: θ t tan =, 2 actuators postons are gven by: where : q = 2 tan b ± ( b ² 4 a c 2 a a = G + M b = 2 N c = G M Usng ths way of solvng equaton (1) a mathematcal sngularty can occur when a =. It s possble to avod ths problem by ntroducng the followng new varables: N G tan α = cos β = M M hen the epresson of q becomes: N G q = tan ± cos M M ² + N ² - KM he set of redundant postons s here trval to determne: r = [ 4θ Note that the multplyng factor (4) comes from the mechancal amplfcaton system nstalled between nacelle and grpper as (Fgure 6). ) u 1 = u y, u2 = u y, u3 = u, u4 = u As t s usual for most parallel robots, the nverse poston relatons s easy to derve. If v s the vector correspondng to the th arm (vector from A to B ), t can be wrtten that ts norm s constant and equal to L: { 1,...,4}, v ² L² = Gears (ato 4 :1) Fgure 6. Amplfcaton system

4 - EJ he frst step to determne EJ s to establsh the relaton between actuators and nacelle velocty respectvely represented by q& and &. As for many parallel robots, ths epresson s derved as follows: J & = J q& (2) o derve equaton (2), the followng classcal property that relates the veloctes V and V B of the two ponts A and B s used: A q 1 M = J 4,1 1 J 4,2 1 J 4, J 4,4 hs lnear over-determned system has to be nverted to get the sought relatonshp that gves the modfed error vector, qˆ. A convenent way to do so s to rely on pseudo-nverson that gves the least square better soluton: VA v = VB v (3) qˆ + = M e If n 1 and n 2 are the vectors from D to C 1 and C 2 respectvely and p the vector from P to B,, the applcaton of equaton (3) to the four robot arms leads to: v1 v2 J = v3 v4 ( p1 v1) ( p2 v2) u y Jq = v1 v1 (n1 v1 v2 v2 (n2 v2 v3 v3 (n2 v3 v4 v4 (n1 v4 ( p3 v3) ( p4 v4) and, f the mechansm s not n a sngular confguraton the jacoban matr s: J = J J q 4. Implementaton and results hs secton presents practcal mplementaton ssues, smulaton and epermental results. Practcal mplementaton: egardng practcal mplementaton, two ssues have been addressed: Œ addng the redundant sensor; an optcal encoder (resoluton: 14 4 tops/rev) mounted on the grpper rotaton as has been selected (Fgure 7, Fgure 8, Fgure 9); t gves a 14 4*4 tops/rev resoluton at nacelle level; Œ nstallng measurement facltes; a 3D vson-based measurement system (AGEI M3D) has been selected; ths system uses 3 lnear CCD fed on the ground (Fgure 1) and a set of 4 LED mounted on the nacelle; a calbraton process has been carred out, but s not descrbed here (the process followed s smlar to the one prescrbed n Fgure 7). 7KXVDZD\IUPVLQJXODUVLWLQVDVPDOOYDULDWLQ Q LV UHODWHG W VPDOO YDULDWLQV Q WKH DFWXDWHG varables, noted q1, L, q4, by the epresson: θ = J 4,1 q1 + J 4,2 q2 + J 4,3 q3 + J 4,4 q4 he relaton between etended errors and jont errors s then: where: M q = e Grpper As Encoder e = [ q 1; q2; q3; q4; θ = [ q 1; q2; q3; q4 Nacelle Fgure 7. he encoder s mounted on the grpper as

5 - a PD jont controller n presence of errors on geometrcal parameters (Fgure 12), - a controller usng the redundant sensor nformaton n presence of errors on geometrcal parameters (Fgure 13). Fgure 8. General vew of H4 equpped wth a redundant sensor, and a 3D measurement system Fgure 11. Control usng a smple PD jont controller (deal case) Fgure 9. Closer vew of the encoder and the 4-LED based measurement system Fgure 12. Control usng a PD jont controller n the case of errors on geometrcal parameters Fgure 1. A vew of the AGEI M3D vson system wth 3 lnear CCD. - mulaton results A Matlab smulaton platform has been mplemented to compare the classcal jont control to the redundant sensor control usng the addtonal measurement of nacelle angle. obot behavor s represented by the dynamc model descrbed n [16. Errors on robot geometrcal parameters are ntroduced, as arms and forearms lengths, motors locatons, nacelle szes: - errors on arms length L from -3mm to 1 mm, - errors on fore arms length l from -.4 mm to.6 mm, - errors on radus from -.5 mm to.5 mm, - errors on d from -.1 to.1 mm. A.2 s dsplacement from = [ mm; mm; -5 mm; rad to f = [1 mm; -1 mm; -4 mm;.2 rad was then smulated n the case of: - a PD jont controller and no error on geometrcal parameters (Fgure 11), Fgure 13. Control usng the redundant nformaton on theta As shown n Fgure 11, when there s no error on geometrcal parameters, a PD jont controller permts to obtan good results (no statc error) whereas t can t ensure a proper control f errors are ntroduced n the model representng the real robot (Fgure 12): a large statc error remans n Cartesan space (.e.: the actuated jonts are at ther desred postons, whle the end-effector locaton s not correct). he nterest of the control usng a sensor measurng nacelle angle s then clearly

6 demonstrated n Fgure 13: statc errors remnd the same on, y and z, but the statfhuuuq LVOZHU - Epermental results he control of H4 robot s realzed by a smple PC (Wndows N, Pentum II 2 MHz) and X (eal me extenson) s used has real tme software to ensure the control task perodcty (see [16 for more nformaton regardng H4 control). In order to compare the results obtaned wth and wthout the redundant sensor, a set of Cartesan postons s obtaned by scannng 625 locatons throughout the H4 workspace (see able 2). Mn. value Ma. value tep sze (mm) y (mm) z (mm) θ (rad) (- ( /12)*4 able 2. Locatons for errors measurement For each locaton, encoders gve motors postons and an estmated nacelle poston s obtaned by computng the FKM. hs estmated poston s then compared to the one gven by the 3D vson based measurement system. he matr M=M(,q) that permts to compute the vector of modfed errors s evaluated for = d and q = q d for not usng the teratve FKM. able 3 sums up the epermental results obtaned for a classcal jont control where a usual controller s mplemented (left value) and when usng the redundant measure of nacelle angle (rght value): $YHUDJH (UUUV PPUGHJ 3HDNW3HDN (UUUV PPUGHJ 6XPI4XDGUDWLF (UUUV PPðGHJð $YHUDJHI $EVOXWH(UUUV PPUGHJ 8VXDO 5HG 8VXDO 5HG 8VXDO 5HG 8VXDO 5HG (UU[ (UU\ (UU (UUθ UP[\ able 3. um-up of the epermental results hese results clearly demonstrate an mprovement of accuracy for the rotaton whereas no change for the others dof (, y, z). he angle bas (.e. the average value of the error) s nearly zero (.2 deg.) n the case of use of the redundant sensor. Moreover, n the same tme, the quadratc error s reduced by 45% (231 vs. 128 mm²). An mportant accuracy mprovement on theta s then realzed, but not to the detrment of errors along, y and z as one could has thought. hs proves that other addtonal sensors could also be added on robot passve jonts and t would ncrease the global accuracy of the robot. 5. Concluson hs paper presented a control strategy based on the use of sensor redundancy for accuracy mprovement. hs method has clearly the advantage of ts mplantaton smplcty and ts low cost (only supplementary sensors are necessary). he frst results obtaned n the case of the H4 robot demonstrated ts valdty wth an ncrease of accuracy when applyng a standard jont space control (t has to be notced that t could be also possble to obtan even better results by mplementng a Cartesan controller usng robot redundant FKM). eferences [1 Gough V.E., Contrbuton to dscusson of papers on research n automotve stablty, control and tyre performance. Proc. Auto Dv. Inst. Mechancal Engneers, [2 tewart D., A platform wth 6 degrees of freedom. Proc. of the Inst. of Mech. engneers, 18(Part 1, 15), pp , [3 Clavel., Une nouvelle structure de manpulateur parallèle pour la robotque légère, APII, 23(6), pp , [4 Merlet J.-P., Les robots parallèles, 2 nd Edton, Hermes, [5 Nahv A., Hollerbach J.M., Hayward V., Calbraton of a parallel robot usng multple knematcs closed loops. In IEEE Int. Conf. on obotcs and Automaton, pp , an Dego, 8-13 May 1994 [6 Lntott A.B. et Dunlop G.., Parallel topology robot calbraton. obotca, 15(4): , 1997 [7 P. Maurne, M. Uchyama, K. Abe, A Fully-Autonomous Procedure for Knematc Calbraton of HEXA Parallel obots, Chna-Japan Blateral ymp. On Adv. Manufacturng Eng., Chna, pp , Oct [8 Zhuang H., Yan J., et Masory O., Calbraton of tewart platforms and other parallel manpulators by mnmzng nverse knematc resduals. J. of obotc ystems, 15(7):395-45, [9 Khall W. et Besnard., elf calbraton of tewart-gough parallel robot wthout etra sensors, IEEE rans. on obotcs and Automaton, 15(6): , 1999 [1 Daney D. et Emrs I.Z., Varable elmnaton for relable parallel robot calbraton. In F.C. Park C.C. Iurascu, edtor, Computatonal Knematcs, pages May 21 [11 K. Cheok, J. Overholt,. Beck, Eact methods for determnng the knematcs of a tewart platform usng addtonal dsplacement sensors, Journal of obotcs ystems, Vol. 1, No 5, pp , 1993 [12 L. Baron, J. Angeles, he sotropc decouplng of the drect knematcs of parallel manpulators under sensor redundancy, Proc. of the IEEE IC&A, pp , [13 I. Bonev, J. yu, N-J. Km, -K. Lee, A smple new closedform soluton of the drect knematcs of parallel manpulators usng three lnear etra sensors, Proc. of the IEEE IC&A, Atlanta, GA, pp , eptember [14 V. Parent-Castell,. D Gregoro, Determnaton of the actual confguraton of the general tewart platform usng only one addtonal sensor, AME Journal of Mechancal Desgn, Vol. 121, pp , March [15 Company O. and Perrot F., A new 3-1 parallel robot, ICA 99, okyo, Japan, October 25-27, 1999, pp [16 Hollerbach, J.M. and Wampler, C.W., he calbraton nde and taonomy for robot knematc calbraton methods. he Internatonal Journal of obotcs esearch, 15(6): , December [17 Perrot F., Marquet F., Company O., Gl., H4 parallel robot: modelng, desgn and prelmnary eperments. IEEE Int. Conf. On obotcs and Automaton, eoul, Korea, May 21 Acknowledgements: hese researches have been partally supported by European Grant GD MACH21.