Numerical model describing optimization of fibres winding process on open and closed frame

Transcription

1 Journal of Physcs: Conference Seres PAPER OPEN ACCESS Numercal model descrbng optmzaton of fbres wndng process on open and closed frame To cte ths artcle: M Petr et al 06 J Phys: Conf Ser Vew the artcle onlne for updates and enhancements Ths content was downloaded from IP address on 07/06/08 at 3:9

2 5th Internatonal Conference on Mathematcal Modelng n Physcal Scences (IC-MSquare 06) IOP Publshng Journal of Physcs: Conference Seres 738 (06) 0094 do:0088/ /738//0094 Numercal model descrbng optmzaton of fbres wndng process on open and closed frame M Petrů, J Mlýnek, T Martnec 3 Deparment of Machnery Constructon, Insttute for Nanomaterals, Advanced Technology and Innovaton, Techncal Unversty of Lberec Lberec, Studentska, 46 7, Czech Republc Department of Mathematcs and Ddactcs of Mathematcs, Faculty of Scence, Humantes and Educaton, Techncal Unversty of Lberec Lberec, Studentska, 46 7, Czech Republc 3 Department of Physcal Measurements, Insttute for Nanomaterals, Advanced Technology and Innovaton, Techncal Unversty of Lberec Lberec, Studentska, 46 7, Czech Republc E-mal: mchalpetru@tulcz Abstract Ths artcle dscusses a numercal model descrbng optmzaton of fbres wndng process on open and closed frame The qualty producton of sad type of composte frame depends prmarly on the correct wndng of fbers on a polyurethane core It s especally needed to ensure the correct angles of the fbers wndng on the polyurethane core and the homogenety of ndvdual wndng layers The artcle descrbes mathematcal model for use an ndustral robot n flament wndng and how to calculate the trajectory of the robot When wndng fbers on the polyurethane core whch s fastened to the robot-end-effector so that durng the wndng process goes through a fbre-processng head on the bass of the sutably determned robot-end-effector trajectory We use the descrbed numercal model and matrx calculus to enumerate the trajectory of the robot-end-effector to determne the desred passage of the frame through the fbre-processng head The calculaton of the trajectory was programmed n the Delph development envronment Relatons of the numercal model are mportant for use a real solvng of the passage of a polyurethane core through fbre-processng head Introducton Compostes are generally lghtweght, hgh strength and flexble, corroson resstant and wth a long lfespan Applyng compostes s sgnfcant especally n the aerospace, arcraft and automotve ndustres The use of ndustral robots n composte producton greatly reduces producton cost, producton tme and mnmzes scrap rate The technology based on a wndng carbon or glass rovng by robot on a non-bearng polyurethane core s currently wdespread n the manufacture of compostes [-5] In ths artcle we focus on the case where the core s formed of a polyurethane frame (whch can be also closed) wth a crcular cross secton An ndustral robot s used n the process of wndng Ths method provdes full control over the placement, the layng drecton, and the amount of fbres on the frame as well as the homogenety of the fbre structure The fnal composte s obtaned after wndng of the flament rovngs on the frame and subsequent thermal curng of the frame The objectve of ths Content from ths work may be used under the terms of the Creatve Commons Attrbuton 30 lcence Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, journal ctaton and DOI Publshed under lcence by IOP Publshng Ltd

3 5th Internatonal Conference on Mathematcal Modelng n Physcal Scences (IC-MSquare 06) IOP Publshng Journal of Physcs: Conference Seres 738 (06) 0094 do:0088/ /738//0094 artcle s to study and mathematcally descrbe the trajectory of the robot-end-effector durng the process wndng of three layers of carbon flament rovngs on a 3D formed frame wth a crcular cross secton The fbre-processng head s fxedly placed n the workspace of the robot and coordnates of ts parts are specfed n the basc coordnate system of the robot The outer gude lnes rotate around a common axs, the ntermedate gude lne s statc The frame s attached to the robot-end-effector The passage of the frame through fbre-processng head s controlled by the movement of the robot-endeffector When polyurethane frame passes through fbre-processng head the strands are successvely wound on the surface of the frame at a targeted angle Frst the outer rotatng gude lne ensures wndng strands under 45 (relatve to the axs of the head and the movng drecton of the frame), subsequently the mddle statc gude wnds the second layer of strands under 0 and the second outer rotaton wnds the fnal layer of strands under -45 Our goal s for the frame to pass through the fbreprocessng head perpendcular to the gude lnes of the head as far as possble Because the poston of the fbre-processng head n the workspace of the robot s fxed and the mddle gude lne s statc the requred poston of the longtudnal fbres on the surface of the wound frame can only be met by sutably turnng the non-bearng frame The prncple of the wndng soluton s shown n Fgure, Fgure Attachng a closed frame to the robot-end-effector Fgure Fbre-processng head wth one (left) and three gude lnes on the frame Numercal Model for descrbe of robot trajectory of wndng fbers on the frame The numercal model for descrbe of robot trajectory of wndng fbers on the frame s see n Fgure 3 Wthn the descrbed numercal model, we wll consder the rght-handed Eucldean coordnate system E3 of the robot () Subsequently, we wll consder the local rght-handed Eucldean coordnate system of the robot-end-effector (LCS) To avod any msunderstandng, we wll label the ponts and vectors wth coordnates n wth the subscrpt and ponts and vectors wth coordnates n LCS wth the subscrpt LCS Actvtes of the ndustral robot are controlled usng a robot control unt (n our case unt KR C4) and the lbrary of nstructons For our purposes t s crucal to set the desred poston of the robot-end-effector usng the lbrary of nstructons The poston and orentaton of the robot-end-effector s defned by usng the LCS The orgn of the LCS s postoned n the robot-endeffector whle at the same tme the robot-end-effector s orented n the drecton of the postve part of the z-axs n the LCS wth regard to the The actual poston of the LCS wth regard to the s determned by the values lsted n the tool-center-pont (TCP) The tool centre pont contans sx values TCP = ( x, y, z, a, b, c) The frst three parameters specfy the coordnates of the orgn of the LCS n regard to the The values a, b and c ndcate the angle of the rotaton of the LCS around the axs z, y and x wth regard to the The non-bearng polyurethane frame can be descrbed by the central axs o see Fgure 4 and radus r TUBE of the cross-secton of the frame The central axs of the frame o s entered n the LCS of the robot-end-effector usng a dscrete set of N ponts stored n the array : array [ N,3],,, B, 3 ndcates the z B where B [ ] ndcates the x, B [ ] ndcates the y and [ ]

4 5th Internatonal Conference on Mathematcal Modelng n Physcal Scences (IC-MSquare 06) IOP Publshng Journal of Physcs: Conference Seres 738 (06) 0094 do:0088/ /738//0094 B [ ] LCS lyng on the axs o, Fgure 4 At the same tme, the array vector B : array [ N,3] s entered, where vector B [, ], vectorb[, ] and vector B [, 3 ] ndcate the coordnates of the unt vector vector B [ ] LCS tangent to the axs o at pont B [ ] LCS In addton, the array vectorb : array [ N,3] s defned, where vector B [, ], vector B [, ] and vector B [, 3 ] ndcate the coordnates of the unt vector vector B [ ] LCS whch when passng the pont B [ ] LCS through coordnate of the -th pont the fbre-processng characterzes the necessary rotaton of the frame about an axs head All the tme vector B vectorb B lyng on the axs [ ] LCS [ ] LCS holds We assume that the dscrete set of ponts [ ] LCS o s specfed suffcently densely and defnes wth a suffcent accuracy the shape of the frame We suppose the dstance of two consecutve pont B[ ] LCS and B[ + ] LCS has a constant dstance h If the polyurethane frame s closed, then the frst entered pont on the axs o s dentcal to the last entered pont ( B[ ] LCS B[ N] LCS, vector B [ ] LCS vectorb [ N ] LCS and vector B [] LCS vectorb [ N] LCS ) The coordnates of the ndvdual components of the fbre-processng head are entered n the coordnate system The frst outer rotatng gude lne s presented by the crcle k wth the = x, y z, the second outer rotatng gude lne s presented by the crcle k wth centre S [ S S, S ] the centre S [ x S, y S, z S ] =, both crcles k and k have the same radus r CIRCLE (we assume S le on the axs s of the fbre-processng head, Fgure 5 r CIRCLE > r TUBE ) Ponts S and Fgure 3 Numercal model for descrbe of robot trajectory of wndng fbers on the frame Fgure 4 Geometry of a polyurethane frame Fgure 5 Ponts S and S le on the axs s of the fbre-processng head The mddle gude lne s statc and enables the placement of the fbres n a longtudnal drecton s not necessary to consder for the model and the subsequent calculatons Pont H s entered, ths pont les n the mddle of the abscssa S S Subsequently, the unt vectors vector H (ths vector ndcates the drecton of passage frame through the fbre-wndng head) and vector vector H are entered, whle the relaton vectorh vectorh s vald We n our test examples selected vector H = = ( S S )/ S S, where S S s the length of the abscssa S S The pont H together wth the vectors vector H and vector H allow us to calculate the passage polyurethane frame through the fbre-processng head whle the frame possbly rotates around the tangent of ts axs o n pont B [ ] 3

5 5th Internatonal Conference on Mathematcal Modelng n Physcal Scences (IC-MSquare 06) IOP Publshng Journal of Physcs: Conference Seres 738 (06) 0094 do:0088/ /738//0094 Mathematcal model of calculatng the trajectory of the robot-end-effector In the dervaton procedure of calculaton of trajectory, we used results lsted n [6-8] Note that the Denavt-Hartenberg method s often used to determne the trajectory of robot [9] Calculaton of the Passage of Frame Thorough the Fbre-processng Head We calculate for =, N the TCP of the robot-end-effector so that at the same tme B[ ] H, vectorb [ ] vectorh, vectorb [ ] vectorh After determnng TCP all of the TCP- parameters are contnuously changed to the parameters of to the TCP By ths procedure we determne partal passage of the frame through the fbre-processng head The ntal TCP ndcates the poston and orentaton of the robot-end-effector before the start of the passage of the frame through fbre-processng head Determnng the Transformaton Matrx We assume that TCP = ( x, y, z, a, b, c ) s entered Wth the am of determnng the transformaton matrx T (whch we wll apply to LCS) we perform the followng steps We determne the coordnates of the vector vector B[ ] LCS n the coordnate system (under the assumpton that the coordnate systems and LCS have the same orgn, ths assumpton s used when fndng the requred rotaton matrx) The matrx Q- of the rotaton LCS towards s n the form Equaton (), (see [], pp 3) Q = Rot(z, a ) Rot(y, b ) Rot(x, c ), () where Rot( z, a ) s the orthogonal matrx of rotaton of LCS around axs z at angle a-, Rot( y, b ) orthogonal matrx of rotaton of LCS around axs y at angle b- and Rot ( x, c ) orthogonal matrx of rotaton of LCS around axs x at angle c- Subsequently, we can express the coordnates of the vector vector B = Q vectorb [ ] [ ] LCS vectorb[ ] LCS By usng the scalar product of vectors vectorh we determne ther devaton α n the system n the form We determne the cross product p = vectorh vector B [] v Vector p s orthogonal to vectors H and vector and vector B [ ] vector B We normalze vector p, e ensure ts unt length p = p / p We perform the rotaton of vector vector B [ ] by angle α around vector p (we consder the correct orentaton of the angle α, e rotate vector vector B [ ] to vector vector H ) Then vectorh = Rot( p, α) vectorb[ ] If we denote p ( ) T = n, n, n3, 0 then the matrx Rot (, α) wll have the form by (), you can see [] p c + n ( c) nn ( c) n3s nn3 ( c) + ns 0 nn ( c) + n3s c + n ( c) nn3( c) ns 0 Rot( p, ) = α () n ( ) ( ) + + ( ) 0 n3 c ns nn3 c ns c n3 c [ ] 4

6 5th Internatonal Conference on Mathematcal Modelng n Physcal Scences (IC-MSquare 06) IOP Publshng Journal of Physcs: Conference Seres 738 (06) 0094 do:0088/ /738//0094 where s and c ndcate s = sn α, c = cosα At the same tme we determne the vector l (3) l = Rot α vectorb[ ] (3) ( p, ) Q The cases where α = 0 or α = π need to be solved separately Q v By usng the scalar product of the vectors we determne the devaton β of vectors vector H and l v The resultng matrx of rotaton Q has the form by (4) ( vector H, ) Rot ( p, α ) Rot( z, a ) Rot( y, b ) Rot( x, c ) β (4) = Rot are analogously defned as elements of the matrx In (4) we use the correctly determned angle orentaton of angle β, e we rotate the vector l (3) around the vector vectorh to the vector vector H where the elements of the matrx Rot ( vectorh, β ) Rot ( p, α ) LCS v The translate vector s defned (5) ( H Q B[ ] ( x y, z, ) T u = 0, (5) LCS, where matrx Q s determned by the relaton (4) We remember, that x, y, z are the frst three parameters of TCP- The resultng transformaton matrx T n the -th step of the passng of the frame through the fbre-processng head s n the form (6) T ( xu ( y, z ) Q = Trans ) where the translate matrx ( x, y, z ) of vector u ( n (5),, (6) 0 0 x 0 0 y Trans = 0 0 and x, y, z are components z After performng of the LCS of the robot-end-effector the correspondng transformaton matrx T determned by relaton (7) wll be vald H B[ ] = T B[ ] LCS and vectorb[ ] vectorh, vectorb[ ] vector H 3 Calculaton of the Euler Angles Any rght-handed rotaton Eucldean space E3 around the gven unt vector p by angle ϕ s determned by the orthogonal matrx Q = Rot ( p, ϕ), ts elements are n the form (3) and det Q = The rotaton matrces form the orthogonal group (5) Each rotaton matrx can be wrtten as a product of the matrces of rotaton around the coordnate axes z, y and x, e Q = Rot( z, a) Rot( y, b) Rot( x, c) (see [], pp 3), where the matrces Rot ( z, a), Rot( y, b) and Rot( x, c) are n the form (); a, b and c are the correspondng Euler angles We note that the determnaton of Euler angles a, b and c s not unque (see [5]) By multplyng the left and rght sde from the left of the matrx Rot ( z, ) because the a a ) and modfed n ths way we can determne the rotaton angles a, b and c (8) When calculatng the angles of rotatons we use the ATAN functon (part of the lbrary of most programmng languages), whch calculates from the two nput parameters arg and arg the value of the functon arctangent for argument arg arg Moreover, the sgns of both matrx Rot ( z, ) s orthogonal Rot ( z, a ) = Rot( z, ) T a 5

7 5th Internatonal Conference on Mathematcal Modelng n Physcal Scences (IC-MSquare 06) IOP Publshng Journal of Physcs: Conference Seres 738 (06) 0094 do:0088/ /738//0094 nput parameters are used to determne the quadrant n whch the resultng value functon s located (t s vald that π < ATAN( arg, arg ) π part) We wrte the matrx of rotaton Q n the form by (7) a = ATAN b = ATAN c = ATAN q ( q ( q3 ( 0 q( q( q3( 0 Q = ( ) ( ) ( ) 0 (7) q3 q3 q ( q(, q( ), ( q3(, q ( cosa + q( sn a ), ( ) q3 ( sn a q3( cosa, q ( cosa q sn a Thus, we determned rotaton angles a, b, c n (8) and the correspondng TCP defnng the poston and orentaton of the LCS of the robot-end-effector n relaton to the of the robot The TCP can be expressed n the form by (9) ( x y, z, a, b c ) TCP =, (9), where the elements of the vector are gven by the relaton (5) (8) 3 Results and summary We focus on the practcal problem of the passage of the non-bearng polyurethane frame wth a crcular cross-secton through the fbre-processng head The central D axs o of the frame s composed of two nterconnected perpendcular arms, Fgure 4 The axs o s entered nto the LCS of the robot-end-effector usng a dscrete set of ponts B, N = 05 The total length d of the [ ] LCS B[ ] LCS to the end pont B [ N ] LCS of the axs s d = 050[ mm] [ ] LCS and [ ] LCS axs o from the startng pont The contnuous dstance of the ponts B [ ] LCS on the axs o from the startng pont s denoted by the varable l The dstance between two consecutve ponts B[ ] LCS and B[ + ] LCS s h = 0[ mm] Vectors vector B vector B are specfed, radus of the frame s r = 0[ mm] The fbreprocessng head s represented by the crcles k and k wth the centres S = [ 000, 05, 990] and S = [ 000, 035, 990] havng the same radus r = CIRCLE 40[mm] S = 70 mm It s vald H = ( S + S ), The length of abscssa S [ ] vector H = ( S S ) S S, vector H = ( 0, 0, ) TUBE As stated n secton IIC, the mddle statc lne enablng the wndng of the fbres to be placed n a longtudnal drecton s not requred for calculatng the trajectory of the robot-end-effector The frame needs to be rotated at a dstance of 550 [mm] from the begnnng of the axs o to the dstance of 690 [mm] around a tangent to the frame of axs o durng the passng of pont B [ ] (for ) by pont H when placng the longtudnal fbres relatve to the axs o of the frame Ths s a unform rght-handed rotary moton wth an overall rotaton angle of ω = π The rotaton s performed durng the passage of the bent porton of the frame through the fbre-processng head The calculaton of the trajectory of the robotend-effector from numercal model referred to n the prevous chapter was appled to the descrbed problem Fgure 6 shows the results of numercal modellng for poston of the robot-end-effector when passng polyurethane frame through wndng head In part a) shows the poston of the robotend-effector (frst three parameters of TCP) and part b) the orentaton of the robot-end-effector (last three parameters of TCP) The descrbed numercal model wth algorthm allows the accurate calculaton and determnaton of the 3D trajectory of the robot-end-effector of the ndustry robot durng the producton of composte profle usng the dry fbre wndng technology on a open and closed non-bearng frame 6

8 5th Internatonal Conference on Mathematcal Modelng n Physcal Scences (IC-MSquare 06) IOP Publshng Journal of Physcs: Conference Seres 738 (06) 0094 do:0088/ /738//0094 a) b) c) d) Fgure 6 Results of numercal modelng of the course of the TCP durng the passng of the frame through the fbre-processng head, a) parameter values of the frst three parameters of TCP, b) parameter values of the last three parameters of TCP, c) trajectory of the robot-end-effector: start and end, d) tme response of numercal model for optmal trajectory of fbres wndng process Acknowledgments The results of ths project No LO0 were obtaned through the fnancal support of the Mnstry of Educaton, Youth and Sports, Czech Republc n the framework of the targeted support of the Natonal Programme for Sustanablty I References [] Shrnzadech B, Alc G, Foog CW and Cassdy G 004 Fabrcaton process of open surfaces by robotc fbre placement, Robotcs and Computer-Integrated Manufacturng, vol 0, pp 7-8 [] Scavcco L and Sclano B 004 Modellng and Control of Robot Manpulators, (London: Sprnger), 004, p 378 [3] Jazar R N 007 Theory and appled robotcs, (New York: Sprnger), p 883 [4] Sevck L, Lepsk P, Petrů M, Mašín I and Martonka R 04 Modern Methods of Constructon Desgn, Lecture Notes n Mechancal Engneerng, (New York: Sprnger), p 63 [5] Petrů M, Kovář R, Martnec T, Srb P, Lufnka A and Kulhavy P 05 Analyss and study of vbratons of a clampng devce used for wndng carbon fbers nto the core of a frame, Proc of the EAN 05-53rd Conference on Expermental Stress Analyss (Cesky Krumlov), pp [6] Martnec T, Mlýnek J, Petrů M 05 Calculaton of the robot trajectory for the optmum drectonal orentaton of fbre placement n the manufacture of composte profle frames, Robotcs and Computer-Integrated Manufacturng, vol 35, pp 4-54 [7] Salabough G G, Computng Euler angles form a rotaton matrx, wwwsoctyacuk/~sbbh635/publcatons/eulerpdf [8] Tan L and Collns C 004 An effectve robot trajectory plannng method usng a genetc algorthm, Mechatroncs, vol 4(5), pp [9] Baločková L 03 The method for solvng knematcs of an ndustral robot, Appled Mechancs and materals, vol 83, pp