Robust algorithm for calibration of robotic manipulator model

Transcription

1 Robust lgorthm for clbrton of robotc mnpultor model Alexndr Klmchk, Yer Wu, Gbrel Abb, Benoît Furet, Antol Pshkevch o cte ths verson: Alexndr Klmchk, Yer Wu, Gbrel Abb, Benoît Furet, Antol Pshkevch. Robust lgorthm for clbrton of robotc mnpultor model. he IFAC Conference on Mnufcturng Modelng, Mngement nd Control (MIM 213), Jun 213, Snt Petersburg, Russ. pp , 213. <hl > HAL Id: hl Submtted on 26 Nov 213 HAL s mult-dscplnry open ccess rchve for the depost nd dssemnton of scentfc reserch documents, whether they re publshed or not. he documents my come from techng nd reserch nsttutons n Frnce or brod, or from publc or prvte reserch centers. L rchve ouverte plurdscplnre HAL, est destnée u dépôt et à l dffuson de documents scentfques de nveu recherche, publés ou non, émnnt des étblssements d ensegnement et de recherche frnçs ou étrngers, des lbortores publcs ou prvés.

2 Robust lgorthm for clbrton of robotc mnpultor model A. Klmchk*, Y. Wu*, G. Abb**, S. Grner***, B. Furet*** nd A. Pshkevch* *Ecole des Mnes de Nntes, 4 rue Alfred-Kstler, Nntes 4437, Frnce; IRCCyN, Nntes 44,Frnce (el: ; e-ml: {lexndr.klmchk, yer.wu, ntol.pshkevch}@mnes-nntes.fr). **École Ntonle d Ingéneurs de Metz, Metz 577, Frnce; LCFC-ENSAM Prsech Metz, (e-ml: bb@enm.fr). ***Unversty of Nntes, Frnce; IRCCyN, Nntes 44,Frnce, (e-ml:{sebsten.grner, benot.furet}@unv-nntes.fr) Abstrct: he pper focuses on the robust dentfcton of geometrcl nd elstosttc prmeters of robotc mnpultor. he mn ttenton s pd to the effcency mprovement of the dentfcton lgorthm. o ncrese the dentfcton ccurcy, t s proposed to pply the weghted lest squre technque tht employs new lgorthm for ssgnng of the weghtng coeffcents. he ltter llows tkng nto ccount vrton of the mesurement system precson n dfferent drectons nd throughout the robot workspce. he dvntges of the proposed pproch re llustrted by n pplcton exmple tht dels wth the elsto-sttc clbrton of ndustrl robot. Keywords: Robot clbrton, robust dentfcton, ndustrl robot, weghted lest squre. 1. INRODUCION he problem of robot clbrton s n the focus of the reserch communty for mny yers (Stone 1987, Hollerbch 1989, Eltt 24). However, there s very lmted number of works tht drectly ddress the ssue of the dentfcton ccurcy nd reducton of the clbrton errors (Moorng 1991, Sun 28, Hollerbch 28). In generl, there exst two mn methods to mprove the dentfcton ccurcy wthout ncresng the number of experments. he frst of them conssts n optmzton of the mnpultor mesurement confgurton used for the clbrton experments. he second method dels wth proper tunng of the dentfcton lgorthms used for estmton of the mnpultor prmeters (fne choosng of the weghtng coeffcents, for nstnt). As follows from the lterture nlyss, the frst method hs been studed n number of ppers (Khll 1991, Borm 1991, Dney 22), whle the second one receved less ttenton n robotc reserch. For ths reson, tkng nto ccount prtculrtes of the mesurement system used n our experments, ths pper focuses on the enhncement of the second method, whch looks rther promsng here. o dentfy the desred prmeters, most of the robot clbrton procedure employ the ordnry lest-squre technque, where ll dentfcton equtons re treted smlrly, wth the sme weghts. hs pproch perfectly suts to the mesurement systems tht provde roughly the sme precson n ll drectons nd n ll workspce ponts. Mthemtclly, t corresponds to the..d.-hypothess concernng the mesurement nose (.e. to the ssumpton tht ll mesurement errors re unbsed, ndependent nd dentclly dstrbuted). However, n ths study, t lest one of these ssumptons s volted becuse the precson of the lser-trcker used n the clbrton experments essentlly depends on the drecton nd the trget mrker locton n the mnpultor workspce. o overcome ths dffculty, the weghted lest-squre technque cn be ppled. As known from lterture, for the lner regresson t gves rther good mprovement nd llows essentlly reducng the mesurement errors mpct. It s evdent tht for the robotc clbrton problem, smlr benefts cn be gned, but the weghtng coeffcent selecton s non-trvl here becuse of the hgh non-lnerty of the equtons descrbng the robotc mnpultor. o ddress ths problem, the remnder of ths pper s orgnzed s follows. Secton 2 defnes the reserch problem nd lso presents some expermentl dt concernng sttstcl propertes of the mesurement errors. In Sectons 3, the dentfcton lgorthm s presented tht s bsed on the weghted lest-squre. Secton 4 dels wth selecton of the weghtng coeffcents tht ensure robustness of the dentfcton lgorthm wth respect to the mesurement nose. Secton 5 presents n pplcton exmple, whch llustrtes benefts of the proposed pproch. And fnlly, Secton 6 summrzes the mn contrbutons of the pper. 2. PROBLEM SAEMEN In robotcs, the clbrton procedure cn be treted s the best fttng of the expermentl dt usng correspondng mnpultor model Π, k rg mn p (,,,, f L q Π k p, ) F (1) whch descrbes ts geometrcl nd elstosttc behvor defned by the known functon f (.) whose prmeters should be tuned. Here p s the vector of mesurements (the Crtesn coordntes of the end-effector trget ponts), the

3 vector L collects ll geometrcl prmeters, q s the vector of ctuted coordntes, the vector Π collects errors n geometrcl prmeters, the vector k collects ll complnces of the mnpultor, the vectors p nd re F used for elstosttc clbrton only nd correspond to the mesurement poston wthout lodng nd ppled externl lodng respectvely. In usul engneerng prctce t s ssumed tht ll mesurements ( nd p p ) re corrupted by the sme mesurement nose N(, 2 ), whch nduces errors ε wth zero expectton E( ε ) nd dgonl covrnce mtrx 2 E( εε ) I. However, for mny mesurement devces such s lser-trckers, the precson hghly depends on the mesurement drecton nd vry throughout the robot workspce. In ths cse, the covrnce mtrx cn be stll ssumed to be dgonl, but wth non-equl prncple elements,.e. E( εε ) dg( x, y, z ), where x, y, z re dfferent nd vry from mesurement to mesurement. hs phenomen s llustrted by expermentl dt presented n ble 1, whch ncludes dspersons of the mesurement errors for the Crtesn coordntes x, y, z for severl mesurement confgurtons used n conventonl clbrton experments. hese results hs been obtned by processng the mesurement dt for 15 confgurtons nd 18 ndependent experments for ech of them (.e. by usng dt set whch conssts of 81 vlues). It should be noted tht the mnpultor repetblty, whch s bout 6μm, does not hve nfluence on the presented results becuse of specfcty of the mesurement experments, where only dfference n the Crtesn coordnte vrtons were evluted (before nd fter pplyng externl lodng). As follows from the presented results, the mesurement error dsperson vry from 17μm to 153μm nd hghly depends both on the drecton (x, y or z) nd the end-effector locton n the mnpultor workspce (correspondng to the confgurtons #1-#15). In prtculr, for the sme confgurton x, y, z cn dffer by the fctor 5. Besdes, from one confgurton to nother correspondng dspersons my dffer by 7 tmes. ble 1. Dspersons of mesurement errors n deflectons for dfferent test confgurtons Confgurton σ x, [m] σ y, [m] σ z, [m] men std men std men std Conf. #1 15 ±1 64 ±1 33 ±1 Conf. #2 57 ±4 86 ±8 118 ±15 Conf. #3 97 ±9 7 ±5 44 ±8 Conf. #4 28 ±1 19 ±1 35 ±1 Conf. #5 72 ±3 48 ±4 17 ±1 Conf. #6 153 ±8 46 ±3 22 ±1 Conf. #7 112 ±6 66 ±3 53 ±4 Conf. #8 74 ±5 55 ±3 59 ±1 Conf. #9 8 ±9 63 ±7 12 ±15 Conf. #1 69 ±2 73 ±1 79 ±1 Conf. #11 8 ±3 36 ±1 26 ±3 Conf. #12 53 ±4 39 ±1 29 ±1 Conf. #13 26 ±1 29 ±1 29 ±1 Conf. #14 88 ±4 121 ±1 42 ±1 Conf. #15 9 ±6 52 ±3 5 ±1 Hence, usul ssumptons ncorportng n robot clbrton technques concernng mesurement nose propertes should be revsed. hs motvtes the prncple gol of the pper tht s med t developng of the robust dentfcton lgorthm tht tkes nto ccount vrtons of x, y, z. 3. IDENIFICAION ALGORIHM In mnpultor geometrc clbrton, the bsc expresson usully wrtten s follows g ( p p J ) ( L, q ) Π (2) g where p s the dfference between the computed v drect geometrcl model nd mesured end-effector poston, the mtrx J s the geometrcl Jcobn nd the superscrpt '(p)' specfes only components tht re relted to the robot poston. It s clerly the lnerzed model, but t s vld here snce n prctce the geometrcl errors re low enough compred to the nomnl vlues of the mnpultor prmeters. Smlrly, for the elstosttc clbrton, the bsc expresson cn be presented n the form e ( p p A ) ( L, q, F) k (3) e where p s the vector of the end-effector dsplcements under the lodng F, the vector k collects ll complnces ( ) of the mnpultor, the mtrx A p defnes the mppng between the jont complnces k nd the nd-effector dsplcements. For the further convenence, both expresson (2), (3) cn be ntegrted n sngle one p B ( L, q, F) X (4) where X Π, k the vector of the unknown prmeters tht should be ( ) dentfed, B p s correspondng mtrx functon. p s the vector of mesurements nd Usng bove defned nottons, the clbrton cn be reduced to the followng optmzton problem m ( ( p ) ) ( ( p ) ) 1 F B Xp B Xp mn (5) tht cn be solved usng the lest squre pproch. It leds to the followng soluton m 1 ( ) ( ) p p m ( p ) 1 1 Xˆ B B B p (6) If the mesurement nose s Gussn (s t s ssumed n conventonl clbrton technques) the covrnce mtrx for the prmeter estmtes ˆX cn be computed s follows m p p 1 1 ) B B where 2 ( ) ( ) m B 1 B s the so-clled nformton mtrx. However, f the mesurement nose vres from confgurton to confgurton the prevous expresson should be revsed. It X (7)

4 cn be proved tht n generl cse the covrnce mtrx s expressed s ) B B B Σ B B B (8) where the mtrx Σ dg( 1, 2,..., 3m ) descrbes the sttstcl propertes of the mesurement errors nd the mtrx ggregtes ll mtrces from expresson (4). B B In order to mprove the dentfcton ccurcy, t s resonble to modfy the objectve functon (5) nd rewrte t n the form m ( ( p ) ) 2 ( ( p ) ) 1 F B Xp W B Xp mn where W dg ( w1, w2,..., w3 m ) s the mtrx of weghtng coeffcents. hs leds to slghtly dfferent expresson for the prmeter estmtes 1 ˆ ( ) p 2 2 X B W B B W p (1) where p ggregtes p. In ths cse, the covrnce mtrx cn be computed s 2 1 ) B W B B W Σ WB B WB 1 X (9) (11) nd, s follows from detled nlyss, the best selecton of the weghtng mtrx corresponds to the equton W Σ I. Hence, to ssgn the weghtng coeffcents, the mesurement nose vrnce should be known. However, n prctce, exct vlues of x, y, z re unknown nd the estmtes cn be used only. On the other hnd, s follows from prctcl experence, smll vrtons n the weghtng coeffcents re not crtcl nd they do not ffect sgnfcntly the dentfcton ccurcy. Nevertheless, f the weghts re ssgned usng smll number of experments, the dentfcton results cn be unpredctbly ffected. herefore, the problem of computng of the weghtng coeffcents tht re ble to ensure robustness of the dentfcton lgorthm s mportnt here nd t wll be n the focus of next Secton. 4. ASSIGNING WEIGHING COEFFICIENS It cn be proved tht for the lner model, the weghng coeffcents ensurng the lowest vrnce of the unknown prmeters cn be computed s w / (12) where s the stndrd devton of the mesurement nose for the -th dentfcton expresson nd the constnt s the scler fctor ntroduced to vod the problem of the unts. For exmple, for the lner regresson wth sngle sclr prmeter, ths pproch llows to reduce the vrnce from ( x )/( x ) to 1/ x / Applyng ths de to the problem of robot clbrton, t s possble to trnsform the covrnce mtrx expresson (11) to the form 2 1 ) B Σ B (13) where ll elements re essentlly lower compred to (8). Hence the problem of nterests s to obtn for ech experment (nd for ech coordnte), whch wll be used for computng weghtng coeffcents w. As t ws mentoned before, computng s bsed on few mesurement my hve the opposte effect - decrese dentfcton ccurcy. But ths remrk does not refer to our cse snce we hve group of ponts (18) n the vcnty of sngle robot confgurton whch llows us to estmte s.t.d. of mesurement nose for x-, y- nd z-drectons wth qute good precson (see ble 1). It should be noted tht obtned vlues of re consderbly hgher thn the clmed ccurcy of the mesurng system (1 μm). Nevertheless, ths vlue cn be used s normlzed coeffcent. Besdes, snce n rel experments t s not possble to be nsured gnst poor dstrbuton of the mesurement errors, n order to ncrese robustness of the dentfcton lgorthm, t s proposed to ntroduce stedy component n the tht n prctce cn be ssgned by the clmed precson of mesurement system 1m. Fnlly. expresson for the weghts tkes form w /( ) (14) where s sclr fctor tht llows to tune the mpct of, s normlzton fctor tht lso llows to vod dvson by zero. 5. APPLICAION EXAMPLE he developed clbrton lgorthm hs been ppled to the ndustrl robot KR-27. o tke nto ccount the compenstor nfluence whle remnng pproch developed for serl robots wthout compenstors (Pshkevch 211), n equvlent vrtul sprng wth non-lner stffness dependng on the jont vrble q 2 s used. Usng ths de, t s convenent to consder severl ndependent prmeters k 2 correspondng to ech vlue of q 2. So, the set of desred prmeters ( k21, k22...), k3,..., k6 cn be denoted s the vector k. hs llows us to obtn lner form of the dentfcton equtons. o fnd optml mesurement confgurtons, the desgn of experments hs been crred out usng ndustry-orented performnce mesure proposed n (Klmchk 212) for fve dfferent ngles q 2 tht re dstrbuted between the jont lmts. For ech q2 three optml mesurement confgurtons hve been found tkng nto ccount physcl constrnts tht re relted to the jont lmts nd the possblty to pply the grvty force (work-cell obstcles nd sfety resons). he results of the clbrton experment desgn re presented n ble 2.

5 ble 2. Mesurement confgurtons Jont ngles, [deg] q 1 q 2 q 3 q 4 q 5 q Grvty compenstor ool Force censor est lodng Fg. 1. Expermentl setup for the dentfcton of the elstosttc prmeters 46 mm ool x F whch gve 81 equtons for dentfcton of 9 desred prmeters. he dentfcton hs been performed usng Ordnry Lest- Squre (OLS) nd Weghted Lest Squre (WLS) technques. For the second pproches the weghts hve been obtned usng non-compensted deflectons fter dentfcton jont complnces, normlzton fctor hs been ssgned to the clmed precesson of the lser trcker (1 m ). Correspondng vlues of the elstosttc prmeters re presented n ble 3, where for WLS the results for.5, 1, 2 re proposed. It lso ncludes the confdence ntervls computed s 3, where the stndrd devton hs been evluted bsed on the expermentl dt usng expressons (8) nd (13). he results show tht the confdence ntervls for OLS nd WLS hve ntersectons for ll prmeters of nterest, moreover, the confdence ntervls for WLS re lwys nsde confdence ntervls for OLS nd consderbly lower (see ble 4). Another mportnt concluson s tht the choce of the coeffcent does not nfluence on the fnl results nd for smplcty cn be ssgned to one. ble 3. Identfed vlues of mnpultor elsto-sttc prmeters usng dfferent pproches, [rd/n m] k OLS WLS =.5 =1. =2. k ± ± ± ±.3 k ± ± ± ±.4 k ± ±.5.32 ±.5.32 ±.5 k ± ± ± ±.1 k ± ± ± ±.7 k ± ± ± ±.11 k ± ± ± ±.71 k ± ± ± ±.12 k ± ± ± ±.267 F 75.5 mm Jont #6 Mrkers Fg. 2. End-effector used for elstosttc clbrton experments o generte elstosttc deflectons, the grvty forces hve been ppled to the robot end-effector (see Fg 1) usng specfc clbrton tool (see Fg. 2). o obtn the desred set of ntl dt, the mnpultor sequentlly pssed through severl mesurement confgurtons. Usng the lser trcker wth the clmed precesson 1 m, the Crtesn coordntes of the reference ponts hve been mesured twce, before nd fter lodng. o ncrese dentfcton ccurcy, three reference ponts (mrkers) hve been used nd the lodng of the mxmum llowed mgntude kg hve been ppled. In ddton, to ensure hgh dentfcton ccurcy for ech confgurton, the experments were repeted sx tmes. In totl the expermentl dt nclude 27 mesurements z It should be noted tht the best weghng coeffcents cn be computed tertvely, where strtng from the second tertons the resduls hve been computed for the elstosttc prmeters dentfed usng WLS wth the weghts obtned on the prevous terton. hs llows us to ncrese ddtonlly dentfcton ccurcy by the fctor 3-2 comprng wth sngle terton WLS. Fnlly, the dentfcton ccurcy hve been ncresed by the fctors nd the errors do not overcome 1.23% for the k 6 nd s.9-.34%. for the remnder prmeters of nterest (such non-equvlent dstrbuton of the dentfcton errors s cused by the specfc of desgn of clbrton experment, where the poston ccurcy fter compenston of the elstc deflectons hs been chosen s performnce mesure (Klmchk 212)). he benefts of WLS re summrzed n ble 4. It ncludes mutul loctons of the results for OLS nd WLS, dentfcton ccurcy nd the benefts of new dentfcton lgorthm comprng wth old one. he convergence of the tertve procedure presented n Fgure 3. Fgure 3 shows vrton n k 21 from terton to terton nd Fgure 3b provdes correspondng Confdence ntervls. hs results shows tht prmeter vlue does not chnge

6 sgnfcntly (3.3% from the orgnl one), however ts precson hs been ncresed by the fctor 4. After 1 tertons the vlues of nd CI re stblzed k 21 ble 4. Benefts of WLS ACKNOWLEDGMEN he work presented n ths pper ws prtlly funded by the ANR, Frnce (Project ANR-21-SEGI-3-2- COROUSSO). he uthors lso thnk Fben ruchet, Gullume Gllot nd Jochm Mrs for ther gret help wth the experments. k Mutul locton of CI CI OLS CI WLS σ OLS /σ WLS k %.9% 4.5 k %.13% 33.2 k %.18% 33.9 k %.33% 33.1 k %.27% 3.1 k 3 7.8%.26% 28.8 k 4 8.2%.25% 35.1 k %.34% 42. k % 1.28% 27.2 k,[rd / N m] k21 CI,[rd / N m] 1 Fg. 3. Convergences of dentfed vlue for k 21 nd ts CI Hence usng new dentfcton lgorthm the elsto-sttc prmeters of the ndustrl robot Kuk KR-27 hve been dentfed wth hgher ccurcy. hs llows us to ncrese the effcency of the error compenston technque proposed REFERENCES Borm J., Menq C. (1991). Determnton of optml mesurement confgurtons for robot clbrton bsed on observblty mesure. Interntonl Journl of Robotcs Reserch, vol. 1, no. 1, pp Dney D. (22). Optml mesurement confgurtons for Gough pltform clbrton. IEEE Interntonl Conference on Robotcs nd Automton (ICRA 22), pp Eltt A.Y., Gen L.P., Zh F.L., Doyun Y., Fe L. (24). An Overvew of Robot Clbrton. Informton echnology Journl, vol. 3, no. 1, pp Hollerbch J. M. (1989). A Survey of Knemtc Clbrton. he Robotcs Revew 1, Cmbrdge, MI Press, pp Hollerbch J.M., Khll W., Guter M. (28). Model Identfcton, Sprnger hndbook of robotcs, Prt B, pp Khll W., Guter M., Enguehrd Ch. (1991). Identfble prmeters nd optmum confgurtons for robots clbrton. Robotc, vol. 9, pp Klmchk A., Wu Y., Pshkevch A., Cro S., Furet B. (212). Optml Selecton of Mesurement Confgurtons for Stffness Model Clbrton of Anthropomorphc Mnpultors. Appled Mechncs nd Mterls, vol. 162, pp n (Klmchk 213). Klmchk A., Pshkevch A., Chblt D., Hovlnd G. (213). Complnce error compenston technque for prllel robots composed of non-perfect serl chns. Robotcs nd Computer-Integrted Mnufcturng Journl, vol. 6. CONCLUSIONS he pper presents robust dentfcton lgorthm for geometrcl nd elstosttc clbrton of robotc mnpultor. In contrst to prevous works, the proposed lgorthm s bsed on the weghted lest-squre technque tht employs new strtegy for ssgnng of the weghtng coeffcents. Such pproch llows us to tke nto ccount vrton of the mesurement system precson n dfferent drectons nd throughout the robot workspce. Applcton of ths technque leds to the essentl mprovement of the dentfcton ccurcy. he dvntges of the proposed pproch re llustrted by n pplcton exmple tht dels wth the elsto-sttc clbrton of n ndustrl robot whch s used for mllng of lrge-dmensonl composte prts. It shows tht the proposed lgorthm llows to reduce most of the dentfcton errors down to.3%, whch s 3-4 tmes better compred to the conventonl technque. 29, ssue 2, pp Moorng B., Drels W.M., Roth Z. (1991). Fundmentls of Mnpultor Clbrton. John Wley & Sons, NY, Pshkevch A., Klmchk A., Chblt D. (211). Enhnced stffness modelng of mnpultors wth pssve jonts. Mechnsm nd Mchne heory, vol. 46, ssue 5, pp Stone H. W. (1987). Knemtc Modelng, Identfcton, nd Centrl of Robot Forwrd Mnpultors, Boston, Kluwer. Sun Y., Hollerbch J.M. (28). Actve robot clbrton lgorthm, IEEE Interntonl Conference on Robotcs nd Automton (ICRA 28), pp