A Path Planning Method for Robotic Belt Surface Grinding

Transcription

1 Chiese Joural of Aeroautics 24 (2011) Cotets lists available at ScieceDirect Chiese Joural of Aeroautics joural homepage: A Path Plaig Method for Robotic Belt Surface Gridig WAG Wei, YU Chao* Robotics Istitute, Beihag Uiversity, Beijig , Chia Received 24 September 2010; revised 25 ovember 2010; accepted 20 December 2010 Abstract The flexible cotact ad machiig with wide strip are two promiet advatages for the robotic belt gridig system, which ca be widely used to improve the surface quality ad machiig efficiecy while fiishig the workpieces with sculptured surfaces. There lacks research o gridig path plaig with the costrait of curvature. With complicated cotact betwee the cotact wheel ad the workpiece, the gridig paths for robot ca be obtaied by the theory of cotact kiematics. The gridig process must satisfy the uiversal demads of the belt gridig techologies, ad the most importat thig is to make the cotact wheel coform to the local geometrical features o the cotact area. For the local surfaces with small curvature, the curve legth betwee the eighborig cuttig locatios becomes loger to esure processig efficiecy. Otherwise, for the local areas with large curvature, the curve legth becomes shorter to esure machiig accuracy. A series of plaes are created to itersect with the target surface to be groud, ad the correspodig sectioal profile curves are obtaied. For each curve, the curve legth betwee the eighborig cuttig poits is optimized by isertig a cutter locatio at the local area with large curvatures. A method of geeratig the gridig paths icludig curve legth spacig optimizatio is set up. The validity is completely approved by the off-lie simulatio, ad durig the gridig experimets with the method, the quality of surface is improved. The path plaig method provides a theoretical support for the smooth ad accuracy path of robotic surface gridig. Keywords: robot programmig; path plaig; belt abrasive; curve legth optimizatio; curvature 1. Itroductio* The automatic millig for workpiece with free surfaces has bee geerally achieved by the computer umerical cotrol (CC) ad other ew maufacturig techologies, but the fiishig process still depeds o experieced maual work. With the rapid developmet of idustry, the complex surface of turbie blades, artificial joits with titaium alloy, saitary ware, digital appliaces ad sport equipmet tur to be more complex ad requirig icreasigly high precisio, which make the traditioal maual machiig borig ad iefficiet. Maual process requires more time ad cost, ad it is iefficiet ad difficult to obtai cosistet *Correspodig author. Tel.: address: cyu18@vip.sia.com Foudatio item: atioal High-tech Research ad Developmet Program of Chia (2007AA04Z2443) Elsevier Ltd. Ope access uder CC BY-C-D licese. doi: /S (11) processig quality, which prevets the process of techological progress. The robotic belt gridig system achieves a high degree of accuracy ad material removal rate i the parts with complex geometries, ad presets a sigificat advace i the state of the arts i surface gridig. Robotic gridig path geeratio becomes oe of the most importat issues to fiish the workpiece with complex surfaces Related work A experieced worker takes several hours to complete the polishig task of a sigle turbie blade, ad the rejectio rate is as high as 15%. Maual gridig operatio is always completed uder poor workig coditios ad high laborig itesity, which seriously threate the health of the operator. Robotic gridig system cosistig of robot system, belt grider, measuremet, calibratio system ad other major subsystems, ca achieve the automatio of fiishig process of workpiece with complex geometries, greatly shorte

2 o.4 WAG Wei et al. / Chiese Joural of Aeroautics 24(2011) the processig cycle, improve processig efficiecy, ad esure the cosistecy of processig quality [1-3]. Robotic belt gridig ca be used to improve the surface quality, part remaufacturig ad part shapig [4]. Robotic belt gridig system with potetial widespread applicatios will replace maual gridig experiece. The fiite elemet model (FEM) betwee wheel ad the local elastic cotact wheel usig the local surface curvature as the geometric boudary coditios was established, ad the depth of cut o the workpiece was further studied ad predicted [5-6]. Ma, et al. poited out that the axis of the cotact wheel must coicide with the pricipal axis of the ormal curvature to achieve the maximum cuttig strip [7]. While the movemet of gridig blades was desiged to obtai the tool path ad umerical code by post-processig, simulatio of the gridig process based o VERICUT CC machie model code was doe. The results are fially applied to practical processig. Shi, et al. cosidered the curvature features of three differet shapes of the cotact wheel, ad proposed the cotrol method of the sixth axis by the priciples of o-iterferece, effective space ad maximum cuttig strip [8]. Huag, et al. preseted a high precisio processig methods of free form surface, established the six-axis gridig ad polishig system, which provided a basis for a reasoable turbie blade polishig process parameters [9]. To uderstad the way of the abrasive grais removig the material i the belt ad workpiece iterface, a method was set up based o the provided importat ad essetial iformatio icludig the ormal load distributio, the local coefficiet of frictio [10]. A ew model based o a eural etwork techique was developed to replace the FEM to calculate the force distributio, which approximated the FEM model with a acceptable tolerace but ca be executed much faster tha FEM [11]. A dual drive curve tool path plaig method was proposed as the tool was positioed alog the dual drive curve uder the coditios of tool path smoothess [12]. The cocept of time-miimal tool paths was itroduced, ad a solutio to the problem of optimal tool path geeratio was give i the form of a itegral equatio [13]. A machiig potetial field method which was costructed by cosiderig both the part geometry ad the cutter geometry to represet the machiig-orieted iformatio o the part surface, was preseted to geerate tool paths for multi-axis sculptured surface machiig [14]. ew algorithms for automatic tool path geeratio for five-axis filleted ed mill fiish-surface machiig were preseted for determiig tool forward step ad tool path step-over that produce a grid-free surface [15]. A special software was developed to pla cuttig path for ruled surface impellers, ad a approximatio algorithm to geerate cuttig path for machiig itegral ruled surface impellers was proposed [16] Motivatio The previous studies regard the ormal vector as a cotact geometric costrait ad calculate the distributio of cotact force. Two pricipal curvature axes should be coicided with the feed directio ad the directio of cotact wheel axis, ad the wide-strip processig ca be achieved. However, the techical requiremets of ormal curvature for the gridig path have ot yet bee deeply researched. O a gridig path composed of a series of discrete cutter locatios, the curve betwee two eighborig cutter locatios is always iterpolated by the lie or circular arc. Durig the gridig process, the desity of cutter locatios is improved dramatically by maual experiece to arrow the curve legth. It avoids the over-cut processig, but it also leads to iefficiet processig. Curvature is the bedig quatitative idicators measurig the path curve i the ormally itersectig plae, so the curve legth betwee adjacet cutter locatios should be arrowed for the cotact area with a large curvature. O the cotrary, it should be icreased. The layout of a robotic belt gridig system is described i Sectio 2. The curve legth optimizatio algorithm for free surface gridig ca be foud i Sectio 3. I Sectio 4, the effectiveess of the algorithm is verified by the simulatio ad gridig experimets. The last sectio presets the coclusio. 2. A Robotic Belt Gridig Workcell A typical robotic belt gridig system ca be show i Fig. 1. With the PPPRRR serial spatial kiematical chai, the gridig robot has a wide space of dexterity ad high cotact stiffess. The workpiece is picked by the ed, while the belt grider stays at a give pose with respect to the base of the robot. As the first three joits are traslatioal, the closed solutio of the iverse kiematics exists. Fig. 1 Robotic belt gridig system. The kiematical schematics ca be show i Fig. 2. The techical parameters of gridig robot ad belt grider are give i Table 1 ad Table 2. The robotic belt gridig system is widely used to grid the sculptured surfaces ad the gridig paths are geerated by the off-lie plaig method.

3 522 WAG Wei et al. / Chiese Joural of Aeroautics 24(2011) o.4 1 Workpiece, 2 Gridig robot, 3 Cotact wheel, 4 Regulatig wheel, 5 Force sesor, 6 Cylider with low frictio, 7 Cylider to tese the belt, 8 Wheel to tese the belt, 9 Belt with abrasive, 10 Liear variable differetial trasformer, 11 Spidle, 12 Assistat wheel, 13 Slider ad rails. Fig. 2 Schematics of robotic belt gridig chai. Table 1 Parameters of gridig robot Item Joit workspace Value X: 400 mm, Y: 400 mm, Z: 500 mm, : 360, : 360, : 360 Load/ 400 Maximum speed/(mms 1 ) 500 Repeatability/mm 0.02 Capacity/kW 4 Table 2 Parameters of belt grider Item Value Rage of cotact force/ Accuracy of force cotrol/ ±10 Adjustig time/ms 25 Belt speed/(ms 1 ) 0-28 Type of cotact wheel Cylider Diameter of cotact wheel/mm Width of cotact wheel/mm Belt legth/mm Capacity of spidle/kw Optimizatio Algorithm to Geerate Gridig Paths 3.1. Problem descriptio The complex geometries are mathematically defied by parametric forms such as the well-kow Bézier, B-splie ad o-uiform ratioal B-splie (URBS) types [17]. The cutter-cotact poits geerated by the method of costat parametric lie, will ot be uiformed. A series of plaes are used to isert with the target surface, ad the correspodig sectioal profile will be obtaied. The cotact wheel will move alog the curves smoothly to cut the metal materials. Without the cosideratio of the ormal curvature, either of the method will be possible to create eough smooth paths. As the dexterity of the gridig robot ad flexible cotact betwee the cotact ad workpiece, robotic belt gridig ca geerally be used to perform the gridig task as the huma labor does. Huma experiece ad aalysis of the geometrical costructio are combied to direct the path geeratio procedures. Before the robot is employed to grid the complex surfaces, the maual gridig procedures must be observed ad recorded. The the feed directios ad style of the arragemet of tool path will be obtaied to get the cutter locatio files. If the dexterity of robotic system has ot passed the evaluatio, some degrees of freedom will be added to the gridig kiematic chai for the curret gridig task. For the gridig task of sculptured surface, the tool paths geerated by the methods of costat parametric lies ad sectioal profiles will ot be applicable. I the robotic belt gridig systems, flexible cotact betwee the cotact wheel ad the workpiece exists, ad a certai amout of materials will be cut by the floatig supportig force ad high belt speed. The cotact stiffess of the robotic belt gridig system turs to be very low, so the cut depth i fact is always less tha the feedig depth. It is obvious that the effective method for decreasig the liear approximatio error is to reduce the step-forward distace ad icrease the umber of cutter locatios [18]. The curve legth of the eighborig cutter cotact must be optimized with the costrait of the local ormal curvatures to guaratee that the cotiuous cuttig depth. Firstly, the cutter locatio files will be obtaied by the differetial formulatios, the the iverse kiematics will be applied to calculatig the robotic joit cofiguratios, which ca be coected by movemet istructios to form a complete robotic gridig path. The procedures of path geeratio ca be explaied as follows: (1) Take the maual procedure as the refereces to obtai the sectioal profile of the gridig paths. (2) Estimate the curve legth of the profile to figure out the correspodig itervals of the gridig cutter cotact poit, ad divide the parametric domai. (3) Calculate the coordiates ad curvature of the cutter locatio o the gridig curve. If the curvature turs to be large eough, the iterval of eighborig cutter locatios o the parametric domai must be arrowed i half by isertig a middle cutter locatio. (4) Calculate the parametric coordiates ad local cotact frames of the gridig cutter cotact poit by the algorithms of surface approachig to obtai the sequeces of the cutter locatios. To obtai the maximum gridig strip width, the X-axis of the local frame is superposed with the miimum pricipal axis of local curvature Path optimizatio Geometrical deductio The defiitio of URBS curve is

4 o.4 WAG Wei et al. / Chiese Joural of Aeroautics 24(2011) C ip, i i i0 0u 1 i0 whose basis fuctio is wp ip, w u u u u i ip1 ip, ip, 1 i1, p1 uip ui uip1ui1 i,0 u i 1 ui u ui1 0 Otherwise (1) (2) (3) whose kot vector is U {0,,0, u 1, p, u,1,,1} (4) p1 p1 p2 where u is the idepedet variable, w i the weight vector of the cotrol poits P i, the umber of the cotrol poits, ad p the order of the URBS curve. The first derivative of the URBS curve is C where (1) (1) ip, ( wp i i) C ip, wi (1) i0 i0 i0 ip, w (1) ip, 1 i 1, p1 ip, p u ip u i u ip1 u i1 The secod derivative of the URBS curve is where C 1 ( ) ( )( ) (2) (2) u ip, u wipi i0 ip, wi i0 i (5) (6) (1) (1) (2) 2 C ip, w i C ip, w i (7) i0 i0 (1) (1) (2) ip, 1 i1, p1 ip, p u ip u i u ip1 u i1 The curvature formula of the curves is (1) (2) C C k= (1) 3 C (8) (9) Whe Eqs. (1)-(8) are substituted ito Eq. (9), the curvature of curve C at u will be calculated. The estimated legth of C is 1 l PP (10) i0 i i1 The defiitio of URBS surface is m i0 j0 m i0 j0 ( v) W ip, jq, ij, ij, S( uv, ) 0 uv, 1 ( v) W P ip, jq, ij, (11) where u ad v are the idepedet variables, W i, j is the weight vector of cotrol poits P i, j, the umber of cotrol poit i u directio, p the B-splie order i u directio, m the umber of cotrol poit i v directio, ad q the B-splie order i v directio. Assume that both S u (u,v) ad S v (u, v) are tagetial vectors of the surface S at (u, v). The ormal vector at (u, v) is ( uv, ) S ( uv, ) S ( uv, ) (12) Let us defie the equatio as u v r( uv, ) S( uv, ) P (13) which refers to the vector from S(u, v) to a certai positio P. To obtai the coordiates (u, v) satisfyig that f( u, v) r( u, v) Su ( u, v) 0 guv (, ) r( uv, ) Sv ( uv, ) 0 (14) ewto-rupso method will be used to calculate the (u, v), which is the parametric coordiates of the cutter locatio. The coverget coditios of Eq. (12) ad Eq. (13) are ( ui+ 1ui ) Su ( ui, vi ) ( vi+ 1vi ) Sv ( ui, vi ) 1 S( ui, vi) P 1 Su( ui, vi) ( S( ui, vi) P) 2 (15) Su( ui, vi) S( ui, vi) P Sv( ui, vi) ( S( ui, vi) P) 2 Sv( ui, vi) S( ui, vi) P where both 1 ad 2 are miimums Algorithm of curve optimizatio Geerally, the curve legth betwee the eighborig cutter locatios o the target surface must be 1-2 mm by the robotic belt gridig system. The cotrol polygo of the sectioal profile curve ca be set up, ad the accurate calculatio of curve legth is time-cosumig ad uecessary. Depedig o the cocave hull property of the URBS curve, the sectioal profile curve will lie iside its cotrol polygo, so the circumferece legth of the cotrol polygo ca be cosidered as the curve legth. The algorithm ca be further explaied as follows: (1) The target surface S, the width b, the diameter d of the cotact wheel ad the allowed curve legth s max of the eighborig cutter locatios are give. (2) The itersectig plae PL i will be created to isert with the target surface S, where the distace of the

5 524 WAG Wei et al. / Chiese Joural of Aeroautics 24(2011) o.4 eighborig plae is less tha b. (3) The sectioal profile curve C i betwee the PL i ad S will be coverted ito a URBS curve. (4) Whe the URBS curve is parsed, the cotrol polygo will be obtaied ad the curve legth l ca be estimated by Eq. (10). Assumig that h=l/s max, a chai of equal-spaced parameters will be set up, {u j u j = u l +(u u u l )/h}u l is the low limitatiou u is the upper limitatio of the parametric domaiad j h. Let j = 0. (5) Usig Eq.(9), the curvature k j of C i o parameter coordiates u j ca be obtaied. (6) If 1/k j <d/4, go to (9); else go to (7). (7) If 1/k j <d/2, go to (8); else a ew cotact wheel with smaller diameter will be substituted ad retur. (8) If j poits to the head of the chai, a ew ode (u j +u j+1 )/2 will be iserted ito the chai betwee odes 0 ad 1, j++; else, a ew ode (u j +u j1 )/2 will be iserted ito the chai betwee odes j ad j1. (9) If the traversal is ot completed, j++ ad go to (5); else, usig Eq. (1), the chai of cutter cotact poit T j will be obtaied, the go to (10). (10) As all the cutter cotact poit of T j are o the surface S, by Eq. (13), the parametric coordiates (s j, t j ) of cutter locatio T j ca be calculated. (11) By Eq. (12), the ormal vector at (s i, t i ) ca be obtaied. (12) Set T j as the origi, the Z-axis, ad the feedig directio as the X-axis, the the local cotact frames of cutter locatios will be obtaied. The retur. The algorithm ca also be explaied i Fig. 3. By iverse kiematics of gridig robots, the gridig paths ca be obtaied. By the ewto s iterative procedure above, a list of discrete robot cofiguratios i correspodece with the list of targets ca be calculated. As the iterval betwee the eighborig cutter cotact poits i the same gridig path is small eough, ad the correspodig eighborig cofiguratios of the gridig robot are iterpolated by the cubic splie method, the ed ca pass the cutter cotact poits smoothly to achieve the robot cotrol programs i the joit space. Fig. 3 Flow chart of path geeratio. 4. Simulatio ad Experimets 4.1. Simulatio by off-lie programmig system The off-lie programmig system of robotic belt gridig system is used to help the programmer geerate the programs of gridig paths, whose core is the graphic simulatio platform. The three modules of kiematics modelig, gridig features ad path geeratio are the base of the graphic simulatio. The off-lie programmig system established ca be show i Fig. 4. The kiematic simulatio of gridig process ca be see i Fig. 5. For a typical gridig task, a curve with uiformed curvatures is show i Fig. 6. The path by the optimizig method i Fig. 7(a) icludes more middle cutter locatios at the local cotact area with large curvatures tha the cotrast path without optimizatio i Fig. 7(b). Fig. 4 Off-lie programmig platform.

6 o.4 WAG Wei et al. / Chiese Joural of Aeroautics 24(2011) Fig. 5 Off-lie simulatio. 5. Coclusios Fig. 8 Cotrast of gridig results. Fig. 6 Curvatures of gridig curve Gridig systems Fig. 7 Path geeratio cotrast. Uder the same gridig parameters, a group of cotrastive belt gridig is doe. If the costat parametric lie method is used to geerate the gridig paths, the surface groud ca be see i Fig. 8(a). The surface gridig quality ca be see i Fig. 8(b) if the algorithm based o the curve iterval optimizatio is used. The smoothess ad accuracy with optimizatio tur to be much better tha those without optimizatio. (1) A belt gridig robot workcell ad off-lie simulatio platform are forwarded. (2) Based o the classical sectioal profile method, a ew algorithm by the optimizatio curve iterval is advaced to geerate more cutter locatios at the local surface with large curvature ad less cutter locatios at the local surface with small curvature. With the optimizatio algorithm, the accuracy ad efficiecy of the robotic belt gridig are itegrated. Firstly, by the cocave hull property, the curve legth of the sectioal profile ca be estimated ad the umber of the curve itervals ca be obtaied. Secodly, the parametric domai of the sectioal profile is divided ito equal itervals, ad parametric coordiates ad positio of the target poits ca be achieved. Thirdly, the curvature is computed poit by poit. If the magitude of the curvature exceeds the give value, a middle cutter locatio will be iserted before the curret cutter locatio. Fially, with the surface iversio, the cutter locatios of the ew gridig paths ca be obtaied. (3) The algorithm is verified ot oly i the off-lie simulatio but also i the robotic belt gridig systems. Refereces [1] Huag H, Gog Z, Che X. SMART robotic system for 3D profile turbie vae repair. Iteratioal Joural of Advaced Maufacturig Techology 2003; 21(4): [2] Su Y, Gibli J, Kazerouia K. Accurate robotic belt gridig of workpieces with complex geometries usig relative calibratio techiques. Robotics ad Computer- Itergrated Maufacturig 2009; 25(1): [3] Whitey D, Brow M. Material removal model ad process plaig for robot gridig. Proceedigs of 17th Iteratioal Symposium o Idustrial Robots. 1987;

7 526 WAG Wei et al. / Chiese Joural of Aeroautics 24(2011) o.4 [4] Wag W, Yu C, Su K. A experimetal method to calibrate the robotic gridig tool. Proceedigs of IEEE Iteratioal Coferece o Automatio ad Logistics. 2008; [5] Re X, Kuhlekotter B. Real-time simulatio ad visualizatio of robotic belt gridig processes. Iteratioal Joural of Advaced Maufacturig Techology 2008; 35(11-12): [6] Zhag X, Kuhlekotter B, Keuper K. A efficiet method for solvig the Sigorii problem i the simulatio of free-form surfaces produced by belt gridig. Iteratioal Joural of Machie Tools & Maufacture 2005; 45(6): [7] Ma Y, Tag X, Che J, et al. The project ad techical study o the track of vertical axis likage abrasive belt gridig of turbie blade. Chia uclear Power 2010; 3(1): [i Chiese] [8] Shi J, Zhag Q. Determiatio of cotact wheel positio ad orietatio for six-axis blade CC abrasive belt gridig system. Mechaical Sciece ad Techology for Aerospace Egieerig 2010; 29(2): [i Chiese] [9] Huag Z, Huag Y, Zhag M, et al. Testig of a six2axis computer umberical cotrol abrasive belt gridig machie based o free-form surfaces. Joural of Chogqig Uiversity 2009; 31(6): [i Chiese] [10] Jourai A, Dursapt M, Hamdia H, et al. Effect of the belt gridig o the surface texture: modelig of the cotact ad abrasive wear. Wear 2005; 259(7-12): [11] Zhag X, Keuper K, Kuhlekotter B. A ew force distributio calculatio model for high-quality productio processes. The Iteratioal Joural of Advaced Maufacture Techology 2006; 27(7-8): [12] Xu R, Che Z, Che W, et al. Dual drive curve tool path plaig method for 5-axis C machiig of sculptured surfaces. Chiese Joural of Aeroautics 2010; 23(4): [13] Radzevich S P. A closed-form solutio to the problem of optimal tool-path geeratio for sculptured surface machiig o multi-axis C machie. Mathematical ad Computer Modellig 2006; 43(3-4): [14] Chiou C, Lee Y. A machiig potetial field approach to tool path geeratio for multi-axis sculptured surface machiig. Computer-Aided Desig 2002; 34(5): [15] Liag Q, Wag Y Z, Fu H Y, et al. Cuttig path plaig for ruled surface impellers. Chiese Joural of Aeroautics 2008; 21(5): [16] Pi J, Red E, Jese G. Grid-free tool path geeratio for five-axis surface machiig. Computer Itegrated Maufacturig Systems 1998; 11(4): [17] Piegl L, Tiller W. The URBS book. 2d ed. ewyork: Spriger-Verlag Berli Heidelberg, [18] Liu X. Five-axis C cylidrical millig of sculptured surfaces. Computer-Aided Desig 1995; 27(12): Biographies: WAG Wei Bor i 1982, he is curretly doig his post doctoral research i Robotics Istitute, Beihag Uiversity. He received his Ph.D. degree from Beihag Uiversity i His mai research iterests iclude automatio i logistics ad robotic maufacturig. jwwx@me.buaa.edu.c YU Chao Bor i 1952, he is curretly a professor i Robotics Istitute, Beihag Uiversity. He received his Ph.D. degree from Tiaji Uiversity i His mai research iterests iclude robotics ad advaced dyamics. cyu18@vip.sia.com