ACTA TECHNICA NAPOCENSIS

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1 573 TECHNICAL UNIVERSITY OF CLUJ-NAPOCA ACTA TECHNICA NAPOCENSIS Series: Applied Mathematics, Mechaics, ad Egieerig Vol. 60, Issue IV, November, 2017 THE ANALYSIS OF GEOMETRICAL ERRORS BASED ON POLYNOMIAL INTERPOLATION FUNCTIONS FOR A 5 D.O.F. SERIAL ROBOT Adia - Veroica CRIȘAN, Floria - Maria ȘERDEAN, Radu MORARIU - GLIGOR Abstract: The paper deals with the experimetal ad umerical modelig of geometrical errors i case of a 5R serial robot (Fauc LR Mate 100iB), implemeted i a workig process which cosists of maipulatig differet parts from the workspace, for the purpose of processig them. For each poit from the workig space covered by the ed effector ad icluded i the aalysis, have bee collected from the teach pedat of the robot a series of data regardig the coordiates of the tool ceter poit, the orietatio agles ad the value of the geeralized coordiates for all drivig joits. The trajectory errors that affect the optimal operatio of the robot have bee determied based o polyomial iterpolatio fuctios of type. Key words: serial robot, dyamic modelig, geometrical errors, polyomial iterpolatio fuctios. 1. INTRODUCTION Robotic idustrial applicatios have experieced a cotiuous developmet sice the first robotic devices have emerged, to owadays. These applicatios also iclude traditioal maufacturig processes but of the late years they have exteded to other iterestig areas such as medical applicatios or etertaiig idustry. Thus, ew capabilities have emerged, like robotic surgery where both the precisio ad the repeatability of robots are essetial. Applicatios based o maufacturig processes have to meet some requiremets which udergo chages depedig o the product type ad quality level, thus imposig robots to adapt to a cotiuous chagig eviromet. The performaces of a robot ca oly be evaluated from the perspective of the applicatios for which it was desiged or i which it is goig to be used [3]. A very importat techological barrier that has affected for a log period of time the robotic idustry, was the problem of reducig or eve elimiatig the error that occurs betwee the work tool referece system ad the referece system attached to the work piece. Forwards are goig to be aalyzed some aspects regardig the compoets of a robotic cell, the referece systems attached, ad the errors that these referece systems are itroducig ito the system [1]. I Fig.1 is represeted a simple robotized cell ad the systems attached to it. Robot S R S C R S FL S TCP Workig space ( robotic cell ) Robot flage The ed effector Workig piece Fig.1. Referece systems for a robotized cell The locatig (positio ad orietatio) of the work piece, is usually defied with respect to the clampig or securig devices, coveyor belts or rotatig platforms, from the robot workspace. From mathematically poit of view, the locatig of the work piece is represeted by a trasformatio that takes place betwee the referece system attached to the work piece ad the system attached to robotic cell. The work piece itroduces a series of errors, mostly due to positio deviatios or deformatios that occur durig the work process [4], [5]. The errors that occur betwee the referece systems attached to the work tool ad the work S P SS P

2 574 piece s referece system is largely due to the differeces that occur betwee the aalytical modelig of the robot ad that of the cotroller, factors such as machiig errors ad moutig, ifluecig i a great deal this type of error [2]. I order to miimize this type of errors, have bee provided differet solutios that just offered a partial resolutio to this problem, such as desigig ad buildig of more rigid structures or rigid fasteers characterized by good repeatability, or improvig the maufacturig processes where the robot is used. Ufortuately, most of the solutios proved to be ofte ufeasible due to the difficult requiremets or just because the costs ivolved were too high. [6] I this paper are preseted some aspects regardig the umerical ad experimetal modelig of locatig errors, based o the trasfer matrix of geometrical errors [8], i case of a five degrees of freedom mechaical robot strucure (Fauc LR Mate 100iB) as well as the aalysis of a workig process which cosists of maipulatig differet parts i the robot workspace for the purpose of processig them. The active compoet of the workig process is the Fauc LR Mate 100iB articulated robot whose performace characteristics are detailed i the followig paragraph. 2. DESCRIPTION OF WORKING PROCESS The idustrial robot, Fauc LR Mate 100iB that was itegrated i the workig process, has bee specially desiged to esure operatio i the idustrial eviromet but it ca be successfully used for didactic purposes i various laboratory applicatios. The Fauc robot has a modular costructio, the coectio betwee the four basic modules beig provided by meas of the five rotatio joits, drive by DC motors. I Fig.2 is represeted the mechaical structure of the Fauc LR Mate 100iB robot, beig highlighted the omial dimesios ad the work space / volume described by the robot i motio. Also, due to the sealed costructio, ecapsulated bearigs, brushless servomotors ad harmoic reducers mouted o all 5 axes, the structure has the advatage of miimizig maiteace, thus icreasig operatig time. As a safety measure, the robot is equipped with fail-safe brakes mouted o axles 2 ad 3 to eable emergecy stoppig whe ay object from the workspace is hit. The robot is recommeded for use i dirty eviromets due to package IP54 that offers protectio agaist dust ad differet types of liquids. Fig. 2 Nomial dimesios ad workig evelope of Fauc LR Mate 100iB The mechaical structure of the robot is desiged to be able to perform loadig ad uloadig operatios of machie tools, high precisio maipulatio operatios, ad whe a work tool is attached to the ed effector, it ca be used for complex weldig operatios requirig high level of precisio. There is also a versio of Fauc LR Mate 100iB robot which is used predomiatly i assembly applicatios that require operatio i clea eviromets at high speeds ad precisio (assembly process of data acquisitio boards, microprocessors, etc.). The robot used i the aalyzed work process has a umber of advatages that recommed it for use i most idustrial applicatios. Rotatio J2 aroud y axis Rotatios J3, J4 aroud y axis Rotatio J5 aroud x axis Rotatio J1 aroud z axis Fig. 3 The mechaical structure of the robot Oe of its mai advatages is the relatively small footprit: 380 x 470 x 320 (mm) which facilitates the istallatio of this robot structure i the immediate proximity of machie tools.

3 575 I Fig. 3 is represeted the mechaical structure of the Fauc type robot, the positio of each drivig joit ad the axis aroud which the rotatio is performed. The Fauc LR Mate 100iB robot structure has bee itegrated ito a complex techological process ivolvig the cooperatio with a mobile platform [K05]. I this paper it is preseted oly the first phase of a complex workig process i which the robot is programmed to start from a iitial cofiguratio ad the to brig the blak i a positio that allows it to be processed. I the ext phases, the processed part is maipulated from the maufacturig cetre ad positioed o the mobile platform from where it is trasported to the warehouse. Aother blak is brought i by the mobile robot ad the process is resumed from the begiig. Hereiafter are preseted the work sequeces cosidered i the aalysis of the geometrical errors, startig with the iitial cofiguratio (Fig. 4) accordig to: p x p y Iitial Cofiguratio p z α z β y Table 1 [ mm ] [ º ] The first phase of the work process that is goig to be aalyzed i the followig cosists of 3 differet sequeces (Fig.4 ad Fig.5) which at their tur are divided ito three trajectory segmets, two ed segmets ad a itermediate segmet thus resultig ito 9 cofiguratios. The study o robot movemet i the workig space was based o the positio ad orietatio data from every drivig joit which were provided by the robot teach - pedat. For each poit covered by ed effector i the workig space ad icluded i the aalysis, a set of data comprisig the coordiates of the tool ceter poit, the orietatio agles, as well as the geeralized coordiates from each robot joit were collected from the robot teach- pedat. The greater the umber of itermediate segmets i which the trajectory is divided, the higher is the accuracy. γ x The purpose of the followig aalysis is to determie the ifluece of geometric errors o the accuracy of the motio trajectory i case of a Fauc serial robot. Fig.4 The robot i the iitial cofiguratio ad Sequece 1 Fig.5 Sequece 2 (left side) ad Sequece 3 (right side) The mechaical structures of the serial ad the mobile robot, as well as the etire workig space were modeled usig Solid Works applicatio. This allowed the gatherig of accurate iformatio regardig the omial values of the geeralized coordiates correspodig to the poits that the robot had to go through durig the workig process. The real value of the geeralized coordiates (comprisig errors) was collected from the robot teach-pedat. k= Iput geometrical errors [ mm ] [ º ] Table 2 p x py pz α z β y γ x I Table 2 are preseted the iput geometrical errors for all 9 cofiguratios take ito study.

4 576 Usig the iformatios provided by the robot teach-pedat were collected the real values (affected by errors) as well as the omial values of the geeralized coordiates from every drivig joit, correspodig to all 9 robot cofiguratios subjected to aalysis. I the table bellow are preseted the values affected by errors of the geeralized coordiates, where ki represets the umber of aalyzed cofiguratios, τ the real time ad tik the time iterval i which the robot reaches the required cofiguratio. k i τ s t ik s Table 3 Geeralized coordiates with errors δ 1jk δ 2jk δ 3jk δ 4jk δ 5jk The geeratio of motio trajectory is doe usig polyomial iterpolatio fuctios, the degree of which depeds o the restrictios imposed by the techological process i which the robot is implemeted. For trajectories of type 4-3-4, the ed segmets are iterpolated with polyomials of 4 th degree ad for the itermediate segmets are used polyomials of 3 rd degree (cubic splie fuctios). 3. THE MODELING OF GEOMETRICAL ERRORS USING THE POLYNOMIAL INTERPOLATION FUNCTIONS Next, the workig process, ivolvig the Fauc LR Mate 100iB is modeled usig polyomial iterpolatio fuctios, thus obtaiig the real iterpolatio fuctios for the error case, based o which is represeted the time variatio of geeralized coordiate errors. The trajectory errors that affect the operatio of the robot i the workig space were determied usig polyomial iterpolatio fuctios. Accordig to the theoretical aspects from the techical literature [7], [8], from this aalysis will result two importat groups of polyomial coefficiets. I the first group are icluded the polyomial coefficiets that are obtaied directly by applyig the iitial coditios i each poit of the motio trajectory. The secod group of polyomial coefficiets is obtaied from a liear ad o-homogeeous system, which is also the result of applyig the cotiuity coditios at all itermediate poits of the motio trajectory. The latter were determied by usig a simulatio program, SimMecRob [6], [8], which allows the umeric ad graphical solvig of the geeralized matrices, of omial kiematics ad dyamics allowig the determiatio of errors for ay structure of serial robot. By usig this simulator SimMecRob were established the omial ad real values of the polyomial coefficiets for all 9 robot cofiguratios subjected to study. Cosiderig the polyomial coefficiets previously determied, the expressios for ormalized polyomial iterpolatio fuctios ca be writte i the followig form: qijk ( t) = aijk4 T k + aijk3 T + aijk2 T + aijk1 T+ aijk0 ; (1) q& ijk ( t) = ( 4 a ijk4 T k + 3 a ijk3 T + 2 a ijk2 T + a ijk1 ) ; (2) && qijk ( t) 1 2 = 2 2 ( 12 a ijk4 T k + 6 a ijk3 T + 2 a ijk2 ) ;(3) where: t; k the first ad itermediate segmet. T = t ; k the last segmet 1; k the ed segmet k =. 0; k itermediate segmet Substitutig i the expressios for the polyomial iterpolatio fuctios the variable τ τk 1 t =, will result the real polyomial iterpolatio fuctios. Based o the polyomial iterpolatio fuctios for omial ad real values, accordig to [6], were determied the errors for the geeralized coordiate, that affect the robot's operatio durig the workig process. For this purpose was computed the differece

5 577 betwee the fuctios real oes that characterize omial cofiguratios ad those that are icludig errors (Tables 4, 5, 6). The polyomial iterpolatio fuctios that characterize the errors from the active drivig joits are preseted i the followig. Segmet Table 4 Polyomial iterpolatio fuctios for errors i the drivig joit q τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ Based o the polyomial fuctios preseted i Table 4, is represeted the time variatio of the drivig joit q 2 as well as its error, for the cosidered phase of the work process. Segmet Polyomial iterpolatio fuctios for errors i the drivig joit q τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ The data cotaied i Table 5 is very useful i establishig the time variatio of the geeralized coordiate q3 ad of its error. Fig.7 The variatio i time of q 3 geeralized coordiate ad of its error for the for the first phase Table 6 Segmet Polyomial iterpolatio fuctios for errors i the drivig joit q 4 Fig.6 The variatio i time of q 2 geeralized coordiate ad of its error for the first phase Table 5 Segmet Polyomial iterpolatio fuctios for errors i the drivig joit q τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ The time variatio of the geeralized coordiate 4 q ad of the errors was obtaied by usig the polyomial iterpolatio fuctios preseted i the Table 6.

6 578 I the fial part of this paper there are preseted i a graphical form the variatio i real time of the geeralized coordiate errors that characterize the studied process ad which were obtaied usig the polyomial fuctios. 5. REFERENCES Fig.8 The variatio i time of q 4 geeralized coordiate ad of its error for the first phase 4. CONCLUSIONS The error that occurs betwee the referece system attached to the work tool ad of the workig part is the effect of the differeces that occur betwee the aalytical modelig of the robot ad that of the cotroller, factors such as machiig or moutig errors havig a great ifluece o this type of error. I order to highlight these errors, it was aalyzed the performace of a Fauc LR Mate 100iB fivedegree serial robot. For this purpose, the robot has bee implemeted i a workig process. Thus, the kiematic commad fuctios for the geeralized coordiates ad their errors could be established, by applyig the polyomial iterpolatio fuctios of type By usig SimMecRob Simulator fuctios, have bee obtaied the expressios of polyomial iterpolatio fuctios for both, omial geeralized coordiates ad real geeralized coordiates (comprisig errors). [1]Borm, J.H, Meg, C.H., Determiatio of Optimal Measuremet Cofiguratios for Robot Calibratio Basesd o Observability Measure. It. Joural of Robotics Research, Vol.10 No.1, [2]Craig, J.J., Itroductio to Robotics, Addiso- Wesley, Amsterdam, [3]Gog, C., Yua, J., No Geometric Error Idetificatio ad Compesatio for Robotic System, It. J Mach. Tools Mauf., [4]Kurfess T.R., Robotics ad automatio hadbook, Edited by Thomas R. Kurfess, CRC Press, [5]Kacso, K., Cotribuții privid modelarea și simularea roboților mobili și roboților cu structură serială, teză de doctorat, febr., [6]Negrea, I., et al, Robotics - Kiematic ad Dyamic Modellig, Editura Didactică şi Pedagogică R.A., ISBN , Bucharest, [7]Negrea, I., Albeţel, D.G., The Geeralized Matrices i the Robot Accuracy, Coferiţa ştiitifică Iteraţioală TMCR 2003, Chişiău, Vol.3, ISBN X, [8]Negrea, I, Duca, A.V, Negrea, D.C., K. Kacso, Mecaica avasată î robotică, Editura UT Press, Cluj-Napoca, ISBN , Aaliza erorilor geometrice bazate pe fuctii poliomoiale petru u robot serial cu 5 GDL Lucrarea are drept scop modelarea experimetală și umerică a erorilor geometrice î cazul uui robot serial 5R (Fauc LR Mate 100iB), implemetat îtr-u proces de lucru care costă di maipularea uor piese di spațiul de lucru, î scopul prelucrării lor. Petru fiecare puct di spațiul de lucru acoperit de efectorul fial și iclus î aaliză, au fost colectate pri itermediul teach-pedatului robotului o serie de date referitoare la coordoatele puctului caracteristic al efectorului fial, ughiurile de orietare și valoarea coordoatelor geeralizate petru toate cuplele active ale robotului. Erorile de traiectorie care afectează fucțioarea optimă a robotului au fost determiate pri aplicarea fucțiilor de iterpolare poliomiale de tip Adia Veroica CRIȘAN, PhD, Lecturer, Techical Uiversity of Cluj - Napoca, Dept. of Mechaical Systems Egieerig, ducaadia@yahoo.com, 0264/ , Floria Maria ȘERDEAN, PhD, Lecturer, Techical Uiversity of Cluj - Napoca, Departmet of Mechaical Systems Egieerig, floria.rusu@omt.utcluj.ro, Radu MORARIU - GLIGOR, PhD, Seior Lecturer, Techical Uiversity of Cluj - Napoca, Dept. of Mechaical Systems Egieerig, rmogli@yahoo.com, 0264/ ,