Trajectory Generation and Control of a 9 axis CNC Micromachining Center

Transcription

1 Trajectory Generation and Control of a 9 axi CNC Micromachining Center A. Yuen, Y. Altinta (1)* Manufacturing Automation Laboratory, Department of Mechanical Engineering, The Univerity of Britih Columbia, Vancouver, BC, V6T 1Z4, Canada A novel 9-axi micro-machine i preented. The machine ha three linear motor driven major axe and a magnetically levitated rotary table with ix degree of freedom. The table can rotate freely with pitch, roll, and yaw and three linear hort troke motion. The major drive bring the tool to the vicinity of target location while the magnetically levitated table poition the machine accurately. All nine drive are controlled imultaneouly by conidering the redundant kinematic of the machine without aturating the troke limit. Simulation and experiment how the generated reference command do not violate the limit of the feed drive, and i ufficiently accurate. Micro machining, Multi-axi, Redundant, Kinematic, Trajectory 1. Introduction There ha been a growing interet in the development of multi-axi micromachining center with preciion poitioning capability due to the miniaturization of engineering component [1] []. Sample application include micro-impeller, micro electromechanical device, urface texturing and micro-die and mold [3]. The conventional machine with erial kinematic configuration lead to accumulated inertia on the main carriage which reduce the bandwidth hence the tracking accuracy. The frictional contact at the drive lead to poor poitioning accuracy, and the ue of air bearing to reduce the friction can be cotly and diturbance tiffne may uffer. It i preferred to have a machine with multiple degree of freedom (DOF) to acce variou part feature with a minimum number of et up, and reduce the frictional contact a much a poible to achieve preciion poitioning at high feed peed. Kinematically redundant machine tool have been introduced to improve the acceible work area and enhance the preciion poitioning with exce drive [4]. Takeuchi et. al [5] uggeted frictionle aerotatic drive to improve the poitioning accuracy, and added a 6 th axi to a 5-axi machine to micro-mill miniature part with complex hape [6]. Lu and Uman preented a frictionle, magnetically levitated high peed table with 6 DOF with a large planar workpace [7]. The ame magnetic levitation principle ha been ued in deigning a rotary table with 6 DOF [8], and integrated to a 3 axi micro-machine to create a 9 DOF micro-machine preented in thi paper. The three axi Carteian gantry machine ha a granite tructure for thermal tability. Although multi-axi machine tool exit in the field of micro-manufacturing, very few have redundant degree of freedom a the configuration preented here. Choi et. al [9] preent a imilar configuration, however it rotation about the C-axi i limited, retricting it poible poitioning capabilitie. The combined ue of the rotary table preented in [8], and it increaed poitioning capabilitie, particularly in the C-axi, make the preented novel 9DOF machine capable of machining miniature feature, free form urface, and urface texture with a variety of micro geometrie. Though it i poible to manufacture complex part with a 5-axi machine, 4 additional axe provide capabilitie uch a geometric and tracking error compenation, or increaing the bandwidth of a redundant axe. The fundamental challenge are how to generate 5 axi tool path trajectory with mooth velocity, acceleration and jerk uing 4 extra, redundant drive without aturating any of the actuator troke, velocity, acceleration, and jerk limit, which are challenge that have not been fully addreed. Though work exit for planning trajectorie with toolpath limit [10] and configuration without redundancy [11] [1], feed planning for redundant configuration ha yet to be addreed. Thi paper preent a novel 9 degree of freedom machine tool including the trajectory generation algorithm and control trategy that allow imultaneou motion on all drive without aturating them. The propoed algorithm are experimentally demontrated on a patial helical path.. Overview of the 9-axi Micro-machine Tool The 9-axi micro-machine tool conit of a main, three axi Carteian machine body carrying a magnetically levitated table with 6 DOF a hown in Figure 1. The main drive are magnetically driven linear motor with x, y, and z peak force of 364N, 580N, and 458N, and continuou force of 91N, 145N, and 73N repectively. Each axi i capable of moving along the Carteian axe x, y, z with a troke of 100mm with an interpolator upported encoder reolution of 40nm. Air bearing upported direct driven Ficher pindle with 00,000 rev/min i mounted on the z axi of the machine. The rotary table i magnetically driven a hown in [8], where the mover conit of a circular Halbach magnet array and the tator i a circular printed circuit board (PCB). Actuation i achieved by paing current through copper coil in the PCB, which create an oppoing magnetic force to the magnet in the mover. The rotary table ha a troke limit of 100μm in the x, y, and z axe, and 0.5 degree in the A and B axe, and unlimited rotation in the C axi. The tranlational axe have a reolution of 0nm while the rotational axe have a reolution of 0.6µrad. Since the table i non-contact, problem aociated with traditional rotary table, uch a friction and backlah, are eliminated [8]. Figure 1. UBC MAL 9-axi micromaching center and the kinematic chain The magnetic actuator i fatened to the top of the -Y axe of the micro-mill a hown in Figure 1. The rotary table i configured o that the tranlational axe of the rotary table are parallel with the tranlational axe of micro-mill. With 3 DOF from the micro-mill and 6 DOF from the rotary table, the micromachining center ha 9 DOF. The micro-machine i controlled by a ingle dspace DS1103 realtime computer. The real-time computer provide three PWM channel and three encoder channel to the micro-mill for current command and poition feedback, repectively. For the rotary table, 4 encoder channel, four analog-to-digital channel, and 8 digital-to-analog channel are required. The 4 encoder channel meaure 4 tranlational location of the rotary table, providing olution for the x, y, and C poition of the rotary table. The 4 analog-to-digital channel interface

2 with 4 capacitance probe that meaure the height of 4 different location of the rotary table, thu olving for the z, A, and B, poition of the rotary table. The 8 digital-to-analog channel are needed to control the 4, three-phae linear amplifier to actuate the rotary table. A a reult, 7 encoder channel, 3 PWM channel, 8 digital-to-analog converter, and 4 analog-to-digital converter are ued to control the whole machine. Since the machine ha more DOF than neceary, a trajectory generation trategy ha been developed o that the generated poition command correpond to the deired tooltip poition and orientation. Furthermore, given the different dynamic of the micro-mill and rotary table, control trategie have been developed for the different drive. The machine i controlled by an in-houe developed, modular, open architecture CNC running on real-time MATLAB upported dspace digital ignal proceing board. 3. Trajectory Generation for the 9-axi Micro-machine Tool The poition of a micro-end mill attached to the pindle can be repreented by it Carteian coordinate P = [P x P y P z ] T and orientation O = [O i O j O k ] T vector with repect to the workpiece. Since the micromachining center ha more DOF than neceary, the relative motion between the tool and workpiece need to be delivered by imultaneouly controlling 9 drive poition, q = [x c y c z c x f y f z f a f b f c f ] T, while reolving the infinite combination of drive poition poible. The ubcript c and f indicate axe on the micro-mill and rotary table, repectively. Furthermore, the poition command mut be cheduled o the limit of the drive are not violated while travering the deired path a fat a poible Forward Kinematic: Given the tool poition (t t p ) and tool orientation (t t o ) in the tool coordinate ytem pecified in NC tool path, the poition and orientation of the tool with repect to the workpiece can be evaluated from the drive poition with the following homogenou tranformation, p P O [ 1 0 ] = H(q) [t o t t t 1 0 ] (1) where H(q) i the coordinate tranformation matrix from the tool coordinate ytem to the workpiece coordinate ytem. The invere of the coordinate tranformation from the bae coordinate ytem to the workpiece i given a: c bf c cf c bf cf b f x c + x f c af cf + c af c cf H(q) b c cf af bf af bf c bf af y cf c + y f w = af cf c cf af + c af c cf bf c af bf c af bf z f cf [ ] where x c y c z c x f y f z f a f b f c f are the drive command of the machine. The trigonometric term are hortened for brevity a follow: c af = co a f, af = in a f, c bf = co b f, bf = in b f, c cf = co c f and cf = in c f. The tranformation from the bae coordinate ytem to the tool: () H(q) b t = [ ] (3) z c where z c i the z axi command of the machine. The tranformation matrix H(q) i evaluated a (Figure 1): H(q) = (H(q) w b ) 1 H(q) t b (4) A a reult, the tooltip poition P(q) and tool axi orientation O(q) are expreed by the poition of all 9 drive pecified in vector (q). 3.. Invere Kinematic: The cutter location command given in the NC program are mapped to the drive uing the invere kinematic, which are olved at the differential level with the Jacobian matrix (J(q)) of the 9 axi micro-machine. The Jacobian (J(q)) i obtained by taking the derivative of P(q) and O(q) from Eq. (1) with repect to the drive poition a hown: J(q) = dp x dp x dx c dc f do k do k [ dx c dc f ] The Jacobian map the differential motion change of the drive to the differential change of the tool poition (dr) a: (5) dr = J(q)dq (6) where R = [P() O()] T i a vector with the deired poition and orientation of the tool a a function of the diplacement () along the toolpath. The tool poition (P()) and orientation (O()) are jerk continuou pline a decribed in [13]. If dr i known, then dq can be found baed on the method propoed in [14]. Firt, the Moore- Penroe peudoinvere of the Jacobian i found a: J (q) = J(q) T (J(q)J(q) T ) 1 (7) Since the deired tool poition and orientation are pecified a a function of diplacement, dr/d can be found analytically. The differential of the drive poition with repect to diplacement dq/d i found from Eq. (7) a [15]: dq d = J (q) dr d α(i J(q)T J (q) T ) φ(q) (8) which lead to many olution of dq/d becaue of extra redundant drive of the machine. Only one feaible olution, which give the lowet motion of all drive to reach target poition without violating the troke limit of the micro-machine and rotary table, need to be identified. The firt term of Eq. (8), J (q)(dr/d) give the minimum average movement of all drive to travel d [16]. The econd term of Eq. (8) guarantee the avoidance of aturating the troke limit of all drive by electing the cot function φ(q) a follow [15]. with the gradient: φ(q) = x c + y c + + c f (9) x c,max φ(q) = [ x c x c,max y c,max y, c y c,max,, c f,max T c f ] c f,max (10) α i a weight factor between the lowet movement of all drive (i.e. firt part of Eq. (8)) and troke limit avoidance (i.e. econd part of Eq. (8)), and mut be iteratively earched to atify all troke limit for all egment along the path. From Eq. (9) and Eq. (10) it can be een that deviation from the center of the drive are penalized. The olution attempt to keep the drive poition centered without violating the troke limit. In hort, all 9 drive move to achieve a 5 axi tool path, without exceeding the hort troke limit of the

3 magnetically rotary table. From the geometric derivative, dq/d, the correponding drive poition can be found by uing a numerical integration cheme uch a the forward Euler integration a follow, q[n + 1] = q[n] + dq[n] d (11) where i a uniform diplacement interval on the toolpath and n indicate the n th interval of command generation. A olution for Eq. (8) and Eq. (11) will alway be poible ince there are no contraint on the geometric derivative The reult of thi algorithm are axe command q, cheduled at fixed toolpath diplacement interval. Thi information can then be ued with a feed optimization trategy which take thi a an input [11] [1] and plan the tangential feedrate, o the limit of the ervo drive are not violated. 4. Control of the 9 drive of the machine The linear motor of the 3-axi micro-mill and the magnetically actuated free floating ma of the 6-axi rotary table are controlled independently a follow: 4.1. Linear Motor Control: The open loop tranfer function between the poition () and applied magnetic force (F) can be modelled by the following rigid body dynamic: F = 1 M + B (1) where M i the ma of the mover and B i the vicou friction from the guideway with roller bearing contact. A proportional poition controller cacaded with a proportional-integral velocity controller i ued to control the linear drive at the initial development of the machine. The cloed loop tranfer function of the axi poition () i derived for a given reference poition command ( R ) a: K p K v + K p K vi = R M 3 + (B + K v ) (13) + (K vi + K p K v ) + K p K vi where K p i the poition controller gain, K v i the velocity controller gain and K vi i the gain of the integral velocity controller. The gain are elected o the velocity loop and poition loop ha a bandwidth of 60Hz and 0Hz, repectively. The performance i further increaed with a feed-forward controller to compenate for the non-linear tatic friction of the drive. 4.. Rotary Table Control: The rotary table i a free floating ma, and it tranfer function between the poition () and applied magnetic force (F) which can be modelled a follow F = 1 (14) m where m i the ma of the mover in the tranlational axe and the moment of inertia in the rotational axe. Since the phae of the tranfer function hown in Eq. i -180 degree, phae ha been added with the following lead-lag controller at the deired bandwidth, C() = F = K (1 + αt 1 + T ) (15) where K adjut the croover frequency, α adjut the amount of phae to be added, and T adjut the frequency at which the phae i added. Thee value are elected to add 60 degree of phae at a croover frequency of 100 Hz, reulting in a bandwidth of approximately 100 Hz. Furthermore, to remove DC error, an integrator (I() = 1 + ω i /) i cacaded with the lead-lag controller. The gain of the integrator, ω i, i elected to be 10Hz o it dynamic do not affect the dynamic of the lead-lag controller. Uing the lead-lag controller and integrator, the axe of the rotary table have the following cloed loop tranfer function: KαT + K(1 + ω i αt) + Kω i = (16) R MT 4 + M 3 + KαT + K(1 + ω i αt) + Kω i 5. Simulation and Experimental Reult Simulation and experiment have been performed to validate the propoed kinematic and control model. The control parameter, ma, and friction of each of the axe are hown in Error! Reference ource not found.. A helical tool path ha been ued to tet the machine. The imulated toolpath and correponding tool poition command are in Figure. The experimental tracking error of all axe are hown in Figure 3 and the accuracy of the machine, a meaured by a laer interferometer, i hown in Table. The tracking error of the micromill axe are within 10μm. The tracking error of the rotary table are 00nm in tranlation and 0μrad in rotation, with the exception of the pike in the C-axi tracking error at.3 econd, where a direction change occur. The propoed trajectory generation algorithm i ued with a feedrate optimization trategy [11] to enure the planned trajectory doe not violate any drive limit. Figure 4 how the computed jerk of the reference command. The jerk hit the limit, indicating the algorithm attempt to optimize the feed profile, but doe not violate the limit. A it can be een, the trajectory generation algorithm can be ued uccefully with exiting optimization trategie to generate movement profile o the 9-axi machine tay within it limit.

4 Table 1. Control parameter of the 9-axi machine Axi PPl Parameter Vicou Inertia K p K v K vi Friction x c kg 7.6 kg y c kg kg z c kg 17.6 kg Lead Lag Parameter Vicou Axi Inertia K Friction x f, y f, z f α = kg.3 kg a T = f, b f kg mm 0 kg mm ω i = 6.83 c f kg mm 0 kg mm Figure 4. Spatial toolpath and poition and orientation command. Figure. Experimental Tracking error of all axe of the micro-machine Table. Laer accuracy tet for 9-axi machine Micromill Accuracy Rotary Table Accuracy x c 3.9 μm x f 0.6 μm y f 0.3 μm y c 5.1 μm z f 3.9 μm a f 0. mrad z c 0.1 μm b f 0. mrad 1.3 mrad 6. Concluion A novel 9-axi micromachining center ha been preented. The 9- axi micromachining center conit of a micro-mill with three linear, large troke axe, and a 6 DOF magnetically actuated rotary table with three linear and three rotational degree of freedom. The fundamental challenge i to deliver a 5 axi tool path by moving all 9 drive with different troke and acceleration limit imultaneouly. The propoed novel geometric trajectory generation algorithm i able to handle the redundant drive without aturating any of the 9 drive. The algorithm optimize the velocitie of redundant drive, i.e. x axi of the mill with a long troke (100mm) and x axi of the rotary table c f Figure 3. Computed jerk of tranlational and rotational reference command with a hort troke (0.mm), o that both move but poition the tool at the deired location. The preliminary deign, kinematic and control method allowed the full operation of the novel micro-machine in real time. The reearcher are currently invetigating new dual-axi control method with improved bandwidth and trajectory generation, while minimizing the diturbance caued by micro-milling force. The metrology of the machine will be developed to compenate it volumetric accuracy. The machine will be ued in micro-end milling of miniature feature and texture. Acknowledgement: Thi reearch ha been upported by National Science and Engineering Council of Canada, Dicovery Grant NSERC Reference [1] Byrne, G., Dornfeld, D., Denkena, B., 003, Advancing Cutting Technology, CIRP Annal - Manufacturing Technology, 5: [] Dornfeld, D., Min, S., Takeuchi, Y., 006, Recent Advance in Mechanical Micromachining, CIRP Annal - Manufacturing Technology, 55: [3] Pier, J., 009, Turbohaft Setup, Internet: [Mar. 1, 016]. [4] Halevi, Y., Carpanzano, E., Montalbano, G., Koren, Y., 011, Minimum energy control of redundant actuation machine tool, CIRP Annal - Manufacturing Technology, 60: [5] Takeuchi, Y., Sakaida, Y., Sawada, K., Sata, T., 000, Development of a 5-Axi Control Ultrapreciion Milling Machine for Micromachining Baed on Non-Friction Servomechanim, CIRP Annal - Manufacturing Technology, 49: [6] Takeuchi, Y., Yoneyama, Y., Ihida, T., Kawai, T., 009, 6-Axi control ultrapreciion microgrooving on culptured urface with non-rotational cutting tool, CIRP Annal - Manufacturing Technology, 58: [7] Lu,., Uman, I.U.-R., 01, 6D direct-drive technology for planar motion tage, CIRP Annal - Manufacturing Technology, 61: [8] Lu,., Dyck, M., Altinta, Y., 015, Magnetically levitated ix degree of freedom rotary table, CIRP Annal - Manufacturing Technology, 64: [9] Choi, Y.-M., Gweon, D.-G., 011, A High-Preciion Dual-Servo Stage Uing Halbach Linear Active Magnetic Bearing, IEEE/ASME Tranaction on Mechatronic, 16: [10] Erkorkmaz, K., Altinta, Y., 001, High peed CNC ytem deign. Part I: jerk limited trajectory generation and quintic pline interpolation, International Journal of Machine Tool and Manufacture, 41: [11] Sencer, B., Altinta, Y., Croft, E., 008, Feed optimization for five-axi CNC machine tool with drive contraint, International Journal of Machine Tool and Manufacture, 48: [1] Dong, J., Ferreira, P.-M., Stori, J.-A., 007, Feed-rate optimization with jerk contraint for generating minimum-time trajectorie, International Journal of Machine Tool and Manufacture, 47: [13] Yuen, A., Zhang, K., Altinta, Y., 013, Smooth trajectory generation for five-axi machine tool, International Journal of Machine Tool and Manufacture, 71: [14] Whitney, D., 1969, Reolved Motion Rate Control of Manipulator and Human Prothee, IEEE Tranaction on Man-Machine Sytem, 10: [15] Liegeoi, A., 1977, Automatic Superviory Control of the Configuration and Behavior of Multibody Mechanim, IEEE Tranaction on Sytem, Man and Cybernetic, 7: [16] Klein, C., Huang, C.-H., 1983, Review of peudoinvere control for ue with kinematically redundant manipulator, IEEE Tranaction on Sytem, Man and Cybernetic, 13:45-50.

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