Maximum Feedrate Interpolator for Multi-axis CNC Machining with Jerk Constraints

Transcription

1 Maximum Feedrate Interpolator for Multi-axi CNC Machining with Jerk Contraint Xavier Beudaert, Sylvain Lavernhe, Chritophe Tournier To cite thi verion: Xavier Beudaert, Sylvain Lavernhe, Chritophe Tournier. Maximum Feedrate Interpolator for Multiaxi CNC Machining with Jerk Contraint. 9th International Conference on High Speed Machining, Mar 2012, San Sebatian, Spain. <hal > HAL Id: hal Submitted on 1 May 2014 HAL i a multi-diciplinary open acce archive for the depoit and diemination of cientific reearch document, whether they are publihed or not. The document may come from teaching and reearch intitution in France or abroad, or from public or private reearch center. L archive ouverte pluridiciplinaire HAL, et detinée au dépôt et à la diffuion de document cientifiue de niveau recherche, publié ou non, émanant de établiement d eneignement et de recherche françai ou étranger, de laboratoire public ou privé.

2 Maximum Feedrate Interpolator for Multi-axi CNC Machining with Jerk Contraint X. Beudaert, S. Lavernhe, C. Tournier LURPA, ENS Cachan, Univerité Pari Sud av du pdt Wilon, Cachan, France Abtract A key role of the CNC i to perform the feedrate interpolation which mean to generate the etpoint for each machine tool axi. The aim of the VPOp algorithm i to make maximum ue of the machine tool repecting both tangential and axi erk on rotary and linear axe. The developed algorithm ue an iterative contraint interection approach. At each ampling period, all the contraint given by each axi are expreed and by interecting all of them the allowable interval for the next point along the path i computed. Example and comparion with an indutrial CNC demontrate the performance of the algorithm. Keyword: Feedrate, CNC, Jerk 1 INTRODUCTION Thi paper deal with the traectory generation which i realized by the Computer Numerical Control (CNC). Indeed, tarting from a NC program the CNC ha to generate the etpoint for the machine' axe. So the input i a reference tool path with a programmed feedrate and the output i a euence of axi etpoint which have to produce a mooth movement. For machining operation, the main goal i to follow preciely a given geometric path, for that reaon, a decoupled approach i preferred. Thi approach conit in decoupling the geometry from the temporal evolution law. The feedrate planning i an old problem which can be olved uing different method and taking into account different contraint. In robotic, Shin and McKay [1] propoed a two pa iterative algorithm baed on the contraint interection principle. The idea i to tart from the beginning of the tool path and to go a fat a poible even if ome contraint are not repected. Then in the revere pa, the procedure i repeated with the additional contraint that the feedrate i lower than in the forward pa. Finally, a corrective algorithm i applied to connect both pae with acceleration contraint. Thi two pa algorithm wa reued and improved. Renton and Elbetawi [2] worked on the velocity and acceleration limit determination. Timar et al. [3] ued polynomial parametric curve on which they could obtained a cloed form olution for the feedrate planning problem with axi acceleration contraint. Dong and Stori [4] tried to prove the optimality of the two pa algorithm. All the previouly cited article took velocity and acceleration contraint of the drive. But it i known that the erk i an important parameter which hould be conidered a well. Indeed in many high peed milling operation the erk i the limiting parameter. The effect of the erk limitation wa tudied in detail by Barre et al. [5]. It i clear that the erk ha to be limited to reduce the freuency content of the traectory and to avoid exciting the natural mode of the tructure. Moreover, everal article are dealing only with the tangential erk (third derivative with repect to the time of the tool/workpiece movement). Thi can be intereting for manipulator but to avoid vibration of a machine tool tructure, each axi limit ha to be conidered. Liu et al. [6] modified the feedrate profile to take into account the erk and the natural freuencie of the machine tool. Thi method hould be applied carefully not to generate contour error. Figure 1: VPOp algorithm Nineth International Conference on HIGH SPEED MACHINING 2012

3 Dong et al. [7] introduced the erk contraint to the two pa iterative algorithm. At firt ight, it doe not eem challenging to add another contraint. But in fact, it i really hard to connect the forward and the backward pa repecting the erk contraint. Indeed a lot of point have to be modified and for complex hape the witching point can be too cloe and the whole feedrate profile ha to be modified by the corrective algorithm. The other main method to realize the feedrate planning with erk contraint i to ue a predefined profile a preented in Erkorkmaz and Altinta [8]. The obective of thi method i to generate a euence of predefined profile which repect the contraint. Thi olution wa alo ued and improved in many other work, e.g. [9-12]. Olabi et al. [13] applied the predefined profile to an indutrial 6-axi machining robot. Lin et al. [14] introduced the control tructure and the contour error in their interpolator. But the main problem i that the contraint on the predefined profile limit only the tangential contraint. In practice each axi ha it own limitation; furthermore with linear and rotary drive it i impoible to make the link between the tangential erk and the axi erk due to the non linear kinematical tranformation. In the literature, few work are realized concerning multiaxi machining with linear and rotary axe. The method with predefined feedrate profile cannot be applied to thi kind of machine a the kinematical tranformation between the carteian pace and the oint pace i nonlinear. Thu the tangential limit have to be really conervative to make ure that the axi kinematical limit are not exceeded. On 5-axi machining with axi erk contraint, Lavernhe et al. [15] preented a predictive model of kinematical performance uing an iterative time invere method. Thi method i uited for G1 tool path but it i difficult to apply it to NURBS tool path. Sencer et al. [16] repreented the feedrate profile with a cubic B-Spline which i iteratively modulated to minimize the machining time. Thi method i uited for NURBS tool path but i difficult to apply to G1 format. The goal of thi paper i to propoe a feedrate planning olution for multi-axi tructure with erk contraint on each axi. A decoupled approach i ued to eparate the geometrical problem to the temporal interpolation problem. Furthermore, the mathematical formulation ued here allow treating both linear and rotary axe in the ame way. Figure 1 how the whole procedure. Firt the geometrical work i carried out. Then the kinematical tranformation i ued to obtain the oint movement. After that, the temporal interpolation i performed uing a contraint interection principle and a dichotomy which will be detailed. Finally the output i the ampled axi etpoint repecting velocity, acceleration and erk contraint of each drive. The plan for the remainder of thi paper i a follow. Section 2 detail the feedrate planning which aim to obtain the velocity profile which make bet ue of the kinematical characteritic of the machine tool. Experiment and imulation are carried out to demontrate the efficiency of the propoed method in Section 3. Finally, the paper i concluded in Section 4. 2 FEEDRATE PLANNING ALGORITHM 2.1 Mathematical formalim and drive contraint Uing the formula for the derivative of the compoition of two function (Euation 1), it i poible to expre the velocity of the drive a a function of the geometry multiplied by a function of the motion. Therefore, the motion i decoupled from the geometry. One can note that thi formula i valid for linear and rotary axe. The acceleration and erk of the drive are obtained identically in Euation 2 and 3. d d d dt d dt (1) ² (2) 3 3,, are the geometrical derivative with repect to the diplacement along the tool path. They are known a oon a the geometrical treatment of the tool path i realized. Becaue of the phyical realization of the drive (motor, driving ytem, machine tool tructure...) the velocity, acceleration and erk of each individual drive have to be limited. The erk limitation i important to reduce the vibration due to the dominating vibratory mode of the axe. Euation 4 preent the velocity contraint. v v v v v max, x max, y max, z max, a max, c X Y Z A C v v v v v max, x max, a max, c (3) max, y max, z (4) All the contraint are et to be ymmetrical a it i commonly ued in the machine tool characteritic. Then the following et of ineuation i obtained repectively for the velocity, acceleration and erk contraint. The notation tand for the abolute value of each calar term. (5) i V ; J i max, i Amax, i; i max, i The aim of the algorithm i to calculate the next reachable point with a fixed Δt knowing all the characteritic on the previou point. Thi i done by interecting all the contraint a propoed in [1]. Uing the dicretization, each contraint can be reduced to a polynomial ineuation. Solving the ineuation, an interval over which the contraint i verified i obtained. Finally, a it i hown on Figure 2, the interection of all thee interval give the olution interval [ min, max] into which all the contraint are repected for thi tep. Thi interval can be empty, for example when the feedrate i too high at the entrance of a harp curvature area. In thi cae, it i neceary to go few tep back and to try to reduce the feedrate a it i hown on Figure 1. It i alo poible to have a olution which i a imple interval or a union of ditinct interval [ min, 2] u [ 3, max]. In thi cae, the minimum and maximum allowable olution min and max are kept.

4 2.2 Detailed algorithm For a dicretized algorithm, two dicretization are conceivable: a geometrical dicretization in Δu or in Δ a in [4] for example or a dicretization in time Δt. The problem with the dicretization in Δ i that eventually you need to end etpoint to the controller with a fixed freuency. So at really low feedrate you need a mall Δ increment to be cloed to the deired command freuency therefore at high feedrate many uele point will be computed. Moreover, with a fixed Δ the evaluation of the erk will be really rough. That i why a contant time tep ha been choen for the algorithm preented here. Hence, the velocity, acceleration and erk are computed a follow: t ;... ; 1 (6) t In order to how explicitly how the algorithm i working, it ha been applied on a really imple example. The tool path i a traight line and the aim i to reach the programmed feedrate tarting from ret. The feedrate will be limited only by the programmed feedrate and the erk of the axi which are repectively 5 m/min and 5 m/ 3. The Figure 3 how each calculated point. On Figure 3.a, one can ee that the algorithm tart going a fat a poible until a point with no olution (empty interection) i reached at =18. Indeed, it i impoible to tay under the programmed feedrate while repecting the acceleration and erk contraint. That mean that the euence ued on the previou point wa wrong. To find the witching point where min ha to be choen intead of max a dichotomy i ued. Figure 2: Contraint interection The idea i to ay that if it i poible to find a euence which allow reaching 0, it will be poible to find a euence which repect all the contraint along any geometry (including any potential upcoming harp corner). To reach 0, it i neceary to decelerate a much a poible o a euence uing alway min i tried. The firt point from where 0 can be reached i find with a dichotomy. On Figure 3.b min i taken from =9 and 0 i reached o it i poible to go further. With =13, 0 i reached again. So on Figure 3.c =15 i teted, no olution i found. The lat iteration of the dichotomy i in =14, here again 0 cannot be reached. So it i ure that 13= min ha to be choen to be able to obtain a olution. Once the dichotomy i finihed, the algorithm trie again to go a fat a poible taking max at each tep. Finally, Figure 3.d how all the point which have been calculated to reach the contant feedrate while repecting the erk contraint. One can ee on Figure 3.e that the feedrate profile i really mooth. Figure 3.f demontrate that the erk limit i repected. The euence of erk preent peak o it i not the one expected mathematically. But a dicretized algorithm will alway create thi kind of olution a the optimal olution i not contained on the et of the dicretized olution. There i no attempt to demontrate the optimality of the olution a it depend on the time tep Δt and on the cheme choen to compute the dicrete derivative. The propoed algorithm i not deigned to be optimal but it i robut and it give a olution which i really cloe to the mathematical optimal olution. A better olution could be obtained by chooing point in the middle of the [ min, max] interval at the expene of complexity and computation load. Figure 3: Detail of the iteration needed to reach a contant feedrate on a traight line

5 3 SIMULATION AND EXPERIMENTAL RESULTS The experiment are carried out on a 5-axi MIKRON UCP 710 machining center (Figure 7 top left) whoe kinematical characteritic are given in Table 1. The machine i controlled by a SIEMENS 840D CNC which allow the meaurement of the poition and velocity of each axi during the movement. The cycle time of the poition control loop i 6 m. The velocity, acceleration and erk of each axi are computed baed on the meaurement of the poition etpoint. It allow to get rid of the noie generated by the mechanical tranmiion which can be een on the linear encoder. To make a fair comparion, the feedrate planning i realized with Δt = 6 m and the derivative are computed in the ame manner uing the derivative cheme of the Euation 6. The computation time with an Intel Core Duo P GHz will be given for information. The algorithm i deigned to be ued off-line but the computation time i atifactory conidering that the algorithm i developed in Matlab environment. V max (m/min - rpm) A max (m/ 2 - rev/ 2 ) J X Y Z A C max (m/ 3 - rev/ 3 ) Table 1: Machine tool drive limit axi native B-Spline machining The bet olution i to ue directly a native polynomial format intead of dicretizing and reinterpolating a et of point. The NURBS format i well adapted for that purpoe and CAM oftware offer the poibility to generate NURBS program. Thi format facilitate the feedrate interpolation proce a there i no need to modify the geometry that i why many reearch paper tart from that format. Here the only thing to do i to find the relation between the path parameter u and the arc length of the curve. programmed feedrate can be reached and harp corner which are difficult to handle. Moreover, a the curve i ymmetric, the feedrate profile hould alo by ymmetric. The Figure 4 how the tool path colored a a function of the feedrate. Obviouly, a it can be een, the feedrate i directly linked to the curvature of the tool path. Thi i clear on a 2D tool path with two euivalent axe, but it will be hown further that it i different in multi-axi machining with linear and rotary drive. The feedrate profile i hown in Figure 5. Here one can ee ome difference with meaured reult on the machine which i not near optimal. Then for the X and Y axi, the velocity, acceleration and erk are computed with the formula given in Euation 6. The reult are preented on Figure 7, for thi example the limiting parameter are the erk of the axe and the programmed feedrate. Furthermore, one can ee that the erk limit i exceeded by the Siemen algorithm. But a thi algorithm i a black box, it i only poible to gue at the reaon of thi behavior. It can be verified that the erk generated by the VPOp algorithm i within the authorized 120% erk overrun limit. The meaured machining time on the Mikron milling machine i 2.4, the planned machining time i 2.3 and the computation time on a PC i 6.2. Figure 5: Trident - Feedrate profile Figure 4: Trident - tool path The example of the third degree trident curve i inpired by the trident ued in [14]. It ha been modified becaue the original uadratic B-Spline wa not C² continuou at the connexion between the arc. Thi example i intereting becaue it contain linear portion where the Figure 6: Trident - Contraint verification

6 3.2 5-axi native B-Spline machining with a 5-axi milling machine and a robot arm To demontrate the generality of the propoed algorithm, the feedrate planning i applied to control the 5-axi Mikron machine and a robot arm with ix degree of freedom. A the application i a milling operation, the lat oint of the robot i redundant with the other o it movement i arbitrarily locked. The kinematical characteritic correpond to thoe of the RX170B machining robot of Stäubli [13] (Figure 7 top right). The native B-Spline tool path decribed on Figure 7 repreent a flank milling operation on an open pocket. To define a 5-axi tool path two curve with the ame parametrization are needed: a bottom curve to define the tool tip traectory and a top curve to define the tool orientation along the curve [17]. The parametrization of the B-Spline doe not matter becaue the tool movement i dicretized according to a near arc length evaluation. Here the only thing to do i to find the relation between the path parameter u and the arc length of the curve. The feedrate diplayed on Figure 7 thank to the colorbar i imulated uing the characteritic of the robot. A it can be een, the feedrate variation do not correpond to the curvature any more. Figure 8 preent the reult of the algorithm and the comparion with the meaurement made on the Mikron machine. The derivative are computed with Euation 6. So it prove that the algorithm repect the erk contraint of each drive. One can ee that the feedrate i mainly limited by the programmed feedrate, velocity and acceleration of the C-axi and erk of X, Y and A axe. The tangential feedrate plot how that the feedrate calculated by the VPOp algorithm i higher than the one meaured on the Siemen 840D. But mot of the time the reult obtained by VPOp are imilar to the one meaured and thi difference i probably due to the real time calculation contraint impoed to the indutrial CNC. The meaured machining time on the Mikron milling machine i 3.7, the planned machining time i 2.8 and the computation time on a PC i 10. Figure 7: Open pocket - 5-axi machine and robot arm. Figure 8: Open pocket - contraint verification on the 5-axi machine.

7 4 CONCLUSION High peed milling reuire mooth traectory generation algorithm able to limit the erk of each linear or rotary axi of the machine tool. Thi paper preent a comprehenive olution which decouple the geometrical treatment of the programmed tool path and the feedrate interpolation. Indeed, if the tool path geometry i not at leat C² continuou the erk limitation cannot be repected. In cae of NURBS tool path decription, the geometry i already continuou o the algorithm ut need to find the relation between the arc length and the parameter. Once a correct geometry i obtained, the feedrate interpolation i performed on the axi movement. Thi iterative algorithm compute the interection of the contraint given by each axi at each tep. Thu a erk limited velocity profile can be obtained for any kind of machine a oon a the kinematical tranformation i known. Several example demontrate that the algorithm i efficient and that the erk of each axi i really repected. Further development could ue thi feedrate interpolator in a 5-axi Open CNC machine tool and allow additional optimization thank to a precie control of the real machining feedrate. All the work preented here i implemented with Matlab and the VPOp oftware can be downloaded from 5 REFERENCES [1] Shin, McKay, 1985, Minimum time control of robotic manipulator with geometric path contraint, IEEE Tranaction on Automatic Control, 30/6: [2] Renton, D., Elbetawi, M. A., 2000, High peed ervo control of multi-axi machine tool, International Journal of Machine Tool and Manufacture, 40/4: [3] Timar, S. D., Farouki, R. T., Smith, T. S., Boyadieff, C. L., 2005, Algorithm for time-optimal control of CNC machine along curved tool path, Robotic and Computer-Integrated Manufacturing 21/1: [4] Dong, J., Stori, J. A., 2006, A generalized timeoptimal bidirectional can algorithm for contrained feed-rate optimization, Journal of Dynamic Sytem, Meaurement, and Control 128: [5] Barre, P.-J., Béarée, R., Borne, P., Dumetz, E., 2005, Influence of a erk controlled movement law on the vibratory behaviour of high-dynamic ytem, Journal of Intelligent and Robotic Sytem 42: [6] Liu, X., Ahmad, F., Yamazaki, K., Mori, M., 2005, Adaptive interpolation cheme for nurb curve with the integration of machining dynamic, International Journal of Machine Tool and Manufacture 45/4-5: [7] Dong, J., Ferreira, P., Stori, J., 2007, Feed-rate optimization with erk contraint for generating minimum-time traectorie, International Journal of Machine Tool and Manufacture, 47/12-13: [8] Erkorkmaz, K., Altinta, Y., 2001, High peed CNC ytem deign part I: erk limited traectory generation and uintic pline interpolation, International Journal of Machine Tool and Manufacture, 41/9: [9] Nam, S.-H., Yang, M.-Y., 2004, A tudy on a generalized parametric interpolator with real-time erk-limited acceleration, Computer-Aided Deign, 36: [10] Lai, J.-Y., Lin, K.-Y., Teng, S.-J., Ueng, W.-D., 2008, On the development of a parametric interpolator with confined chord error, feedrate, acceleration and erk, The International Journal of Advanced Manufacturing Technology 37/1-2: [11] Heng, M., Erkorkmaz, K., 2010, Deign of a NURBS interpolator with minimal feed fluctuation and continuou feed modulation capability, International Journal of Machine Tool and Manufacture, 50/3: [12] Lee, A.-C., Lin, M.-T., Pan, Y.-R., Lin, W.-Y., 2011, The feedrate cheduling of NURBS interpolator for CNC machine tool, Computer-Aided Deign, 43/6: [13] Olabi, A., Béarée, R., Gibaru, O., Damak, M., 2010, Feedrate planning for machining with indutrial ixaxi robot, Control Engineering Practice, 18/5: [14] Lin, M.-T., Tai, M.-S., Yau, H.-T., 2007, Development of a dynamic-baed NURBS interpolator with real-time look-ahead algorithm, International Journal of Machine Tool and Manufacture 47/15: [15] Lavernhe, S., Tournier, C., Lartigue, C., 2008, Kinematical performance prediction in multi-axi machining for proce planning optimization, The International Journal of Advanced Manufacturing Technology, 37/5-6: [16] Sencer, B., Altinta, Y., Croft, E., 2008, Feed optimization for five-axi CNC machine tool with drive contraint, International Journal of Machine Tool and Manufacture, 48/7-8: [17] Langeron, J. M., Duc, E., Lartigue, C., Bourdet, P., 2004, A new format for 5-axi tool path computation, uing bpline curve, Computer-Aided Deign 36/12: