Precision Control of High Speed Ball Screw Drives

Transcription

1 Preciion Control of High Speed Ball Screw Drive by Amin Kamalzadeh A thei preented to the Univerity of Waterloo in fulfillment of the thei requirement for the degree of Doctor of Philoophy in Mechanical Engineering Waterloo, Ontario, Canada, 8 Amin Kamalzadeh 8

2 AUTHOR'S DECLARATION I hereby declare that I am the ole author of thi thei. Thi i a true copy of the thei, including any required final reviion, a accepted by my examiner. I undertand that my thei may be made electronically available to the public. ii

3 Abtract Indutrial demand for higher productivity rate and more tringent part tolerance require fater production machine that can produce, aemble, or manipulate part at higher peed and with better accuracy than ever before. In a majority of production machine, uch a machine tool, ball crew drive are ued a the primary motion delivery mechanim due to their reaonably high accuracy, high mechanical tiffne, and low cot. Thi bring the motivation for the reearch in thi thei, which ha been to develop new control technique that can achieve high bandwidth near the tructural frequencie of ball crew drive, and alo compenate for variou imperfection in their motion delivery, o that better tool poitioning accuracy can be achieved at high peed. A preciion ball crew drive ha been deigned and built for thi tudy. Detailed dynamic modeling and identification ha been performed, conidering rigid body dynamic, nonlinear friction, torque ripple, axial and torional vibration, lead error, and elatic deformation. Adaptive Sliding Mode Controller (ASMC) i deigned baed on the rigid body dynamic and notch filter are ued to attenuate the effect of tructural reonance. Feedforward friction compenation i alo added to improve the tracking accuracy at velocity reveral. A bandwidth of 3 Hz wa achieved while controlling the rotational motion of the ball crew, leading to a ervo error equivalent to.6 um of tranlational motion. The motor and mechanical torque ripple were alo modeled and compenated in the control law. Thi improved the motion moothne and accuracy, epecially at low peed and low control bandwidth. The performance improvement wa alo noticeable when higher peed and control bandwidth were ued. By adding on the torque ripple compenation, the rotational tracking accuracy wa improved to.95 um while executing feed motion with m/ec velocity and g acceleration. A one of the main contribution in thi thei, the dynamic of the t axial mode (at 3 Hz) were actively compenated uing ASMC, which reulted in a command tracking bandwidth of 8 Hz. The mode compenating ASMC (MC-ASMC) wa alo hown to improve the dynamic tiffne of the drive ytem, around the axial reonance, by injecting additional damping at thi mode. After compenating for the lead error a well, a tranlational tracking accuracy of.6 um wa realized while executing m/ec feed motion with.5 g acceleration tranient. In term of bandwidth, peed, and accuracy, thee reult urpa the performance of mot ball crew driven machine tool by 4-5 time. iii

4 A the econd main contribution in thi thei, the elatic deformation (ED) of the ball crew drive were modeled and compenated uing a robut trategy. The robutne originate from uing the real-time feedback control ignal to monitor the effect of any potential perturbation on the load ide, uch a ma variation or cutting force, which can lead to additional elatic deformation. In experimental reult, it i hown that thi compenation cheme can accurately etimate and correct for the elatic deformation, even when there i 3% variation in the tranlating table ma. The ED compenation trategy ha reulted in 4. um of tranlational accuracy while executing at m/ec feed motion with.5 g acceleration tranient, without uing a linear encoder. Thi reult i epecially ignificant for low-cot CNC (Computer Numerically Controlled) machine tool that have only rotary encoder on their motor. Such machine can benefit from the ignificant accuracy improvement provided by thi compenation cheme, without the need for an additional linear encoder. iv

5 Acknowledgement I would like to expre my incere gratitude to my upervior, Dr. Kaan Erkorkmaz, for the upport and knowledge he ha provided to me during my graduate tudie. It ha been a pleaure and a great experience working with him. I would alo like to acknowledge my friend and colleague at the Preciion Control Laboratory (PCL), a well a the faculty and taff member of the Univerity of Waterloo, Mechanical and Mechatronic Engineering Department. In particular, I would like to thank Dr. Amir Khajepour for hi upport and guidance, and Mr. Robert Wagner and Mr. Andy Barber for their help in deign and manufacturing of the experimental etup. I would like to thank Mr. Wilon Wong, my good friend and colleague from PCL, who ha helped me in the early tage of real-time oftware coding and intrumentation. Finally, I greatly appreciate the encouragement, upport, and love of my dear family, which truly made my graduate tudie poible. v

6 Dedication I dedicate thi thei to Nahal Doroudian who accompanied me along the way and provided upport, encouragement, and acrifice. I deeply appreciate all her help and upport. vi

7 Table of Content Lit of Figure... x Lit of Table... xiv Chapter Introduction... Chapter Literature Review Modeling of Ball Screw Drive Rigid Body Dynamic Friction Structural Vibration Motion Delivery in the Preloaded Nut Control Law Deign Tracking veru Contouring Control Controller Deign for High Accuracy and Vibration Avoidance Compenation of Torque/Force Ripple Compenation of Elatic Deformation Concluion... 5 Chapter 3 Modeling and Identification Introduction Experimental Setup Rigid Body Dynamic Identification Theoretical Inertia Calculation Leat Square Identification of Inertia, Vicou, and Coulomb Friction Identification of the Amplifier Current Loop Detailed Identification of Friction Characteritic uing a Kalman Filter Vibratory Dynamic Finite Element Modeling of Torional Mode and Experimental FRF Validation Modeling and Meaurement of the Firt Axial Mode Concluion Chapter 4 Adaptive Sliding Mode Controller Deign Introduction General Formulation of Adaptive Sliding Mode Control (ASMC) ASMC Deign for Rigid Body Dynamic... 4 vii

8 4.4 Notch Filtering of Structural Reonance Notch Filtering of Torional Mode Notch Filtering of the t Axial Mode Nonlinear Friction Compenation High Speed Tracking Tet Rotational Feedback Uing Linear Feedback Limitation Concluion Chapter 5 Modeling and Compenation of Torque Ripple Introduction Identification of Torque Ripple uing a Kalman Filter Torque Ripple Compenation Experimental Tracking Reult Limitation Concluion... 6 Chapter 6 Active Compenation of Axial Vibration Introduction ASMC Deign for a Flexible Drive Sytem Drive Model Conidering the Firt Axial Mode ASMC Formulation State Trajectory Generation Modeling and Compenation of Lead Error Experimental Reult Frequency Repone Meaurement High Speed Tracking Reult Stability Analyi Limitation Concluion... 8 Chapter 7 Modeling and Compenation of Elatic Deformation Introduction Modeling of Elatic Deformation and Lead Error viii

9 7.. Lead Error and Backlah Elatic Deformation Elatic Deformation Compenation in Feedforward Elatic Deformation Compenation in Feedback Formulation of Feedback ED Compenation ED Model Identification Integration into the Control Scheme Stability Analyi High Speed Tracking Reult Uing only Rotary Feedback Uing Combined Rotary and Linear Feedback Limitation Concluion... Chapter 8 Concluion Concluion Future Reearch Concluion... 4 Bibliography...5 Appendice: Appendix A : Leat Square Identification of Rigid Body Dynamic... Appendix B : Kalman Filter Deign... 5 Appendix C : Finite Element Analyi of the Ball Screw Drive... 3 ix

10 Lit of Figure Figure.: Meaured (olid line) and predicted (dahed line) tranfer function of ball crew drive (ource: Varanai and Nayfeh [68])... 7 Figure.: Hyterei behaviour of the ball crew nut (ource: Cuttino et al. []): (a) Angular diplacement hyterei, (b) Friction torque hyterei Figure 3.: Experimental etup Figure 3.: Rigid body dynamic model Figure 3.3: Current amplifier and motor block diagram.... Figure 3.4: Meaured current loop frequency repone function.... Figure 3.5: Kalman filtering... 3 Figure 3.6: Etimated diturbance with Kalman filtering, jogging at a peed of mm/ec Figure 3.7: Oberved and curve fit friction model Figure 3.8: The finite element model in ANSYS Figure 3.9: Torional mode: (a) t mode, (b) nd mode, (c) 3rd mode Figure 3.: Meaured open loop acceleration FRF Figure 3.: Amplifier, motor, and ball crew block diagram Figure 3.: Acceleration meaurement at 4 Hz Figure 3.3: Torional FRF with different amplitude of input ignal and the table at the middle poition. Meaurement from: (a) Encoder, (b) Encoder, (c) Encoder Figure 3.4: Torional FRF with different table poition. Meaurement from: (a) Encoder, (b) Encoder, (c) Encoder Figure 3.5: Experimental and curve-fit torional FRF for: (a) Encoder, (b) Encoder, (c) Encoder Figure 3.6: Axial and Torional FRF (Rotary encoder refer to Encoder ) Figure 3.7: Axial vibration FRF at different table location Figure 4.: t order linear liding urface Figure 4.: Original and notch-filtered torional FRF, meaured from: (a) Encoder, (b) Encoder Figure 4.3: Original and notch filtered FRF of the t axial mode Figure 4.4: Deigned ASMC with notch filtering and feedforward friction compenation (poition feedback loop cloed with Encoder ) Figure 4.5: Commanded poition trajectory x

11 Figure 4.6: Cloed loop tracking FRF without and with active damping of the t torional mode.. 47 Figure 4.7: Experimental high peed tracking reult for Adaptive SMC uing rotary feedback, with notch filtering Figure 4.8: Experimental high peed tracking reult for Adaptive SMC uing rotary feedback, without notch filtering Figure 4.9: Propoed control trategy for active vibration uppreion of the t torional mode Figure 4.: Contribution of active damping in attenuating ball crew torional vibration Figure 4.: Deigned ASMC with notch filtering of the torional and axial mode and feedforward friction compenation (Linear encoder ued for feedback) Figure 4.: Experimental high peed tracking reult for ASMC uing linear feedback Figure 5.: Diturbance obervation uing a Kalman filter Figure 5.: Torque ripple harmonic Figure 5.3: Oberved and modeled torque ripple amplitude and phae hift varying with: (a) Axi velocity, (b) Torque command Figure 5.4: Modeled and oberved diturbance at different axi velocitie Figure 5.5: Adaptive SMC with torque ripple compenation block diagram Figure 5.6: Experimental tracking reult at mm/ec uing ASMC with torque ripple compenation Figure 5.7: Experimental tracking performance without and with torque ripple compenation for: (a) Low bandwidth ASMC, (b) High bandwidth ASMC Figure 6.: Open loop acceleration FRF Figure 6.: Flexible drive model ued in controller deign Figure 6.3: Meaured and modeled lead error profile Figure 6.4: Removal of lead error from the control loop for ingle axial feedback cae Figure 6.5: Removal of lead error from the control loop for dual (rotary-axial) feedback cae... 7 Figure 6.6: Mode compenating ASMC tructure Figure 6.7: Experimental FRF for linear poitioning Figure 6.8: Diturbance tranfer function meaured by hammer tet Figure 6.9: Tracking performance of MC-ASMC without lead error correction Figure 6.: Tracking performance of MC-ASMC with lead error correction Figure 6.: MC-ASMC block diagram: (a) Feedforward and feedback tranfer function, (b) Rearranged to obtain the loop tranfer function xi

12 Figure 6.: Nyquit diagram of mode compenating ASMC ued in tracking experiment Figure 6.3: Stability of MC-ASMC: Marginally table cae. (a) Nyquit prediction, (b) Tracking error, (c) Control ignal Figure 6.4: Stability of MC-ASMC: At the tability margin. (a) Nyquit prediction, (b) Tracking error, (c) Control ignal Figure 6.5: Stability of MC ASMC: Marginally untable cae. (a) Nyquit prediction, (b) Tracking Error, (c) Control Signal Figure 6.6: Stability of MC-ASMC: Untable cae. (a) Nyquit prediction, (b) Tracking error, (c) Control ignal Figure 7.: Schematic of ball crew for modeling elatic deformation Figure 7.: Meaured and modeled lead error Figure 7.3: Feedforward (open loop) elatic deformation compenation Figure 7.4: Tuning of elatic deformation model parameter Figure 7.5: Feedback (cloed-loop) elatic deformation compenation Figure 7.6: Derivation of the loop tranfer function for Nyquit tability analyi.... Figure 7.7: Variation of loop tability with the choice of ED compenation filter: (a) Effect of filter order, (b) Effect of corner frequency. Shaded region indicate the influence of poition dependent drive flexibility.... Figure 7.8: No ED or LE compenation, only rotary feedback, nominal table ma Figure 7.9: No ED or LE compenation, only rotary feedback, 3% increaed table ma... 5 Figure 7.: Feedforward (open loop) ED and LE compenation, only rotary feedback, nominal table ma Figure 7.: Feedforward (open loop) ED and LE compenation, only rotary feedback, 3% increaed table ma Figure 7.: LE and feedback (cloed-loop) ED compenation, only rotary feedback, nominal table ma Figure 7.3: LE and feedback (cloed-loop) ED compenation, only rotary feedback, 3% increaed table ma... 6 Figure 7.4: No ED or LE compenation, combined rotary and linear feedback, nominal table ma Figure 7.5: No ED or LE compenation, combined rotary and linear feedback, 3% increaed table ma xii

13 Figure 7.6: Only LE compenation, combined rotary and linear feedback, nominal table ma, max. er.: 6. um, rm:.58 um Figure 7.7: Only LE compenation, combined rotary and linear feedback, 3% increaed table ma, max. er.: 9.8 um, rm:.95 um Figure 7.8: LE and feedback (cloed-loop) ED compenation, combined rotary and linear feedback, nominal table ma Figure 7.9: LE and feedback (cloed-loop) ED compenation, combined rotary and linear feedback, 3% increaed table ma xiii

14 Lit of Table Table 3.: Inertia parameter for the ball crew drive ytem... 9 Table 3.: Identified parameter.... Table 3.3: Pole, zero, and gain of 3rd order approximation of the current loop.... Table 3.4: Pole, zero, and gain of 6 th order torional FRF oberved by Encoder Table 3.5: Pole, zero, and gain of 6 th order torional FRF oberved by Encoder Table 3.6: Pole, zero, and gain of 6 th order torional FRF oberved by Encoder Table 4.: Torional notch filter parameter, uing T = /, ec for dicretization Table 5.: Identified amplitude and phae value for torque ripple harmonic Table 7.: Experimental data ued in contructing the lead error offet look-up table Table 7.: Summary of maximum and RMS (root mean quare) tracking error value xiv

15 Chapter Introduction Ball crew drive are currently the mot common mean of delivering high preciion motion in work machine, uch a machine tool, where both high rigidity and poitioning accuracy are required. They have everal ditinctive advantage uch a low cot, high mechanical tiffne, large diplacement troke, and the ability to provide robutne to the ervo ytem againt work (e.g. cutting) force and load inertia variation due to their inherent gear ratio. However, becaue of their contact-type deign ball crew drive are ubject to wear. They are alo limited by lower acceleration and velocity value, compared to direct drive (e.g. linear motor), which are gradually becoming more accepted in the machine tool indutry. However, linear motor are ignificantly more expenive and their control i more challenging, due to the lack of a motion tranmiion ratio. Thi reult in the diturbance or load variation to be directly felt by the motor, thereby bringing eriou robutne iue. Hence, a tried and teted technology, ball crew drive are till in widepread ue in machine tool a well a other type of production machinery. In fact, major machine tool builder continue to invet in and maintain their own dedicated production line to manufacture the ball crew drive required for their machine. The challenge aociated with controlling any type of feed drive ytem, whether it i ball crewdriven or direct drive baed, are achieving high poitioning accuracy at elevated peed and acceleration; maintaining a ufficient amount of tiffne over a wide frequency range for diturbance force rejection; and delivering a pecified performance in a robut manner in the preence of acceptable variation in the feed drive dynamic. With the recent advance in high peed machining, maintaining the dynamic tool poitioning accuracy ha become more important than ever before, in order to be able to take advantage of the productivity gain facilitated by high cutting peed [57]. Thi in return bring new challenge in term of control law deign. High dynamic accuracy require the poitioning bandwidth to be greater than even before [53], in order to follow the rapidly varying tool command without violating the part geometric tolerance. The trend i now toward new motion control technique that can achieve reponive frequency range (i.e. bandwidth) beyond Hz. Thi require higher ampling frequencie and finer poition feedback to be ued; both of which are available through more powerful real-time computer and high reolution poition enor (i.e. inuoidal encoder or laer interferometer). In addition, it i vital to have a profound undertanding of the dynamic that govern the repone of a feed drive ytem, o that adequate compenator can be deigned that take full advantage of it phyical capabilitie. Thi i

16 eential for achieving the highet poible dynamic accuracy and diturbance rejection characteritic. Thi thei follow a ytematic approach in modeling the dynamic of ball crew drive and introduce new control technique that deliver higher motion accuracy at elevated peed. All modeling and control work i carried out on a high preciion ball crew drive, upported on air guideway, that wa built at the Preciion Control Laboratory at the Univerity of Waterloo. The etup i intrumented with multiple poition enor at different location, which allow detailed phyical model and new control technique to be developed and experimentally validated. Henceforth, the proceeding chapter in thi thei are organized a follow: A literature review on the exiting modeling and control technique for ball crew drive i preented in Chapter. In particular, the high order dynamic compriing of axial and torional vibration i of interet, a one of the goal in thi thei i to develop new control technique that can achieve bandwidth in the vicinity of thee tructural mode. In addition, imperfection of the ball crew motion delivery originating from lead error and motion lo in the preloaded nut are alo of concern. Other iue that are reviewed are the rigid body dynamic, friction characteritic, torque ripple, thermal expanion, and elatic deformation. Compenation trategie that deal with uch effect are alo urveyed. Baic dynamic of the ball crew drive are modeled and identified in Chapter 3, compriing of rigid body motion, current loop dynamic, nonlinear friction characteritic, a well a the torional and axial vibration. The vibration mode are modeled uing Finite Element and analytical approache. They are validated experimentally with frequency repone meaurement. The model and parameter identified in thi chapter are ued in the following chapter for controller deign and tability analye. In Chapter 4, the baic control methodology choen in thi thei, which i Adaptive Sliding Mode Control (ASMC), i introduced and ued for controlling the rigid body dynamic. The general ASMC formulation, developed by Slotine and Li [6] and further improved by Zhu et al. [75], i preented. A a pecial cae, the control of a rigid body baed plant with an unknown external diturbance i tudied. Thi cae boil down to the well-known PID controller with feedforward acceleration and velocity compenation term. However, the ASMC deign provide an efficient model-baed mean of tuning the feedback and feedforward gain. In order to improve the ervo tracking accuracy at motion reveral, feedforward friction compenation i applied. In addition, to avoid exciting the tructural reonance through the control ignal, notch filter are deigned for the torional and axial

17 mode. Two feedback cenario are tudied: the firt one uing only rotational meaurement from the ball crew, and the econd one uing direct tranlational poition feedback obtained from the table. Rotary feedback reult in a collocated control ituation, where high bandwidth can be achieved with minimal interference from the t axial and t torional mode (repectively at 3 Hz and 445 Hz). However, the tranlational accuracy of the drive i not guaranteed. In thi cenario, active cancellation of the ball crew torional vibration i alo invetigated. The econd cae with direct tranlational feedback, on the other hand, bring a ignificant limitation in term of the achievable control bandwidth (<7 Hz). Thi i due to the non-collocated control ituation it caue in term of the t axial mode. In Chapter 5, the force/torque ripple in the motion delivery, which can occur in both ball crew a well a direct drive ytem, are identified, modeled, and compenated in the control law. The torque ripple model comprie of the larget harmonic component, which are identified uing a Kalman filter [3] for diturbance obervation. It i hown that compenating for the torque ripple can provide a ignificant improvement in the poitioning accuracy and motion moothne of feed drive, particularly during low peed movement and when there are limitation on the achievable control bandwidth. In Chapter 6, the ASMC i extended to actively compenate for the dynamic of the t axial mode, reulting in the o-called Mode Compenating ASMC (MC-ASMC). Thi i one of the major contribution in thi thei. The MC-ASMC reult in uperior poitioning accuracy over the rigid body baed deign which wa implemented with notch filtering of the t axial mode. The achieved command tracking bandwidth i 8 Hz, which i 4-5 time higher than the bandwidth realized in ball crew driven CNC (Computer Numerically Controlled) machine tool. The MC-ASMC alo improve the damping and diturbance rejection characteritic around the t axial mode, indicating a more favorable repone in term of avoiding machining chatter vibration [3]. In conjunction with active vibration damping, the lead error in the ball crew drive have alo been modeled and compenated, which yield a further improvement in the dynamic poitioning accuracy. With thi developed cheme, a tranlational tracking accuracy of.6 um ha been maintained while travering at mm/ec peed with.5 g acceleration tranient. Thi reult urpae the performance of mot ball crew-driven CNC machine tool by 4-5 time in term of accuracy and peed. Although the MC- ASMC wa demontrated to give highly promiing reult, during it practical implementation it wa een that tuning thi controller wa not a trivial tak. Hence, in order to aid in the deign and afe implementation of MC-ASMC, a detailed tability analyi i conducted and the tability prediction are verified in further tracking experiment, at the end of Chapter 6. 3

18 In Chapter 7, a different approach i taken for realizing high poitioning accuracy and tiffne. Intead of compenating for the axial vibration, the quai-tatic elatic deformation (ED) i etimated and cancelled out in the control law. A implified lumped model i ued for predicting the torque that i tranmitted through the preloaded nut. Thi model i alo capable of capturing the drive poitiondependent flexibility characteritic. Both feedforward and feedback compenation technique have been invetigated. The feedback baed elatic deformation compenation, which i the econd major contribution in thi thei, i hown to be robut againt dynamic change (e.g. ma variation) on the load ide. Thi i becaue thi approach continuouly monitor the real-time control ignal, which help it detect and account for uch perturbation. Thi compenation trategy reult in an additional feedback loop, for which the tability implication have been tudied by conducting Nyquit analye. It i hown that the propoed feedback baed ED compenation trategy alway reult in an improvement in the drive tranlational accuracy. Thi i particularly ignificant for low-cot ball crew drive that have only rotational feedback, typically on the motor haft. The performance improvement i alo demontrated when both linear and rotational feedback are available. During the coure of thi reearch, all of the developed control algorithm were validated in tracking experiment, a well a in frequency repone command following and diturbance rejection (i.e. impact hammer) tet. Currently the ball crew etup i undergoing retrofit o that thee algorithm can alo be validated in more realitic machining experiment in the near future. 4

19 Chapter Literature Review The growing demand for high productivity manufacturing have motivated reearch in new control technique for multi-axi machine. In order to produce, aemble, or manipulate part in minimum time without violating their tolerance, multi-axi machine are deigned to achieve controlled highpeed and high-accuracy movement. With the recent advance in high peed machining [57], pindle peed and material removal rate have increaed by an order of magnitude in the lat two decade. Typical high-peed machine tool today have pindle peed around 3, to 4, rpm. To take advantage of uch high cutting peed, the feed drive alo need to provide fat tool or part motion, while maintaining or improving the poitioning accuracy. Thi require the ervo ytem to have a high bandwidth [53] and good diturbance rejection characteritic. Thi i required to track udden change in the commanded poition trajectory, while being robut againt external diturbance uch a machining force. A another requirement, mooth trajectory generation i alo eential in order to contrain the motion command in the low frequency range. Thi help to avoid exceive tracking error and alo prevent exciting the machine tructural vibration, contributing to the overall machined part quality [][6]. Thi chapter review ome of the reearch on dynamic modeling and control of ball crew drive. Section. addree the work modeling and identification. Section. focue on controller deign. The concluion are preented in Section.3.. Modeling of Ball Screw Drive Dynamic modeling of ball crew drive can be performed at different complexity level conidering variou effect uch a the rigid body motion, bearing and guideway friction, electrical dynamic, torional, axial, and bending flexibility, high order vibration, hyterei type motion lo in the preloaded nut, and motor and mechanical torque ripple. Rigid body modeling i one of the implet approache, which conider the effect of inertia a well a vicou and Coulomb type friction. More advanced modeling that conider torional, axial, and poibly bending vibration [49][68][7], the kinematic and dynamic that influence the motion lo in the preloaded nut [][59][7], the velocity dependent nonlinear friction [5][7][38], and motor torque ripple [45][46][5], enable more accurate prediction of the drive repone. In return, thi knowledge can be incorporated into control 5

20 law deign to achieve better tracking and diturbance rejection characteritic. Some of the reearch on modeling of ball crew drive i preented in the following... Rigid Body Dynamic Rigid body dynamic capture the fundamental behavior which dominate the low frequency range. Thi eentially conit of rigid inertia, vicou damping, and Coulomb friction effect. Rigid body dynamic identification wa performed by Erkorkmaz and Altinta [7] by uing the Leat Square [44] approach. In thi method, an input ignal (u ) i provided to the motor, which i proportional to motor torque in the low frequency range, when the motor i operating in current control mode. The motor torque can be expreed in the form τ = K a K t u where K a and K t are the current amplifier gain and motor torque contant, repectively. By applying a erie of piecewie-contant input and uing the Leat Square technique on the logged velocity and motor torque data, the inertia and vicou damping parameter can be identified conidering the following rigid body dynamic model: ω( ) = [ τ Td ] J B Above, ω () i the drive velocity expreed in the Laplace (i.e. ) domain. J i the inertia, and B i the vicou damping. T d i the diturbance impoed by contant Coulomb friction oppoing the direction of travel. T can be repreented with two parameter ( T and d identification, where T d = T when ω > and T d = T when ω >. (.) T ) in the Leat Square Thi technique i ued for identifying the rigid body dynamic of the ball crew in Section More detail of the rigid body identification technique can be found in [7] and Appendix A... Friction Friction i one of the mot ignificant ource of diturbance in CNC (Computer Numerical Controlled) machine. It typically caue tracking error pike during motion reveral, which occur at harp corner or circular arc quadrant in the toolpath. Thi i due to a udden and dicontinuou change of the true friction value at a zero velocity croing, to which the controller diturbance adaptation law (or integral action) cannot immediately repond. Hence, accurate identification and pre-compenation of friction help alleviate the correponding poitioning error at motion reveral. Dedicated friction modeling and compenation technique that conider the Stribeck effect [5] where tatic friction change to dynamic friction, ha been preented in literature [5][7][38]. 6

21 Identification of friction can be performed in variou way, among which the ue of Kalman filtering [3] to oberve the equivalent diturbance ha been adopted in thi thei following the work in [7]. Thi approach wa found to be effective in capturing the Stribeck curve accurately. Among compenation technique, feedforward friction compenation wa found to be more effective compared to the feedback approach, a it did not interference with the cloed loop dynamic, which can have it own tability implication...3 Structural Vibration Structural vibration comprie an eential part of the ball crew drive dynamic. Thee vibration, if not adequately dampened out or avoided, can limit the achievable control bandwidth to be ignificantly below the t vibration mode [54]. Accurate knowledge of the vibration mode i neceary for the application of mot vibration compenation technique. Structural vibration have been modeled in literature through analytical method [8][68], frequency repone meaurement [][6][68] and Finite Element (FE) modeling [][][49][6][7]. With knowledge of the vibration mode, feedforward (FF) compenation technique can be developed which avoid exciting the torional or axial vibration of ball crew drive, uch a the one preented by Chen and Tluty [8]. In feedback uppreion technique, a lumped model can be ued to repreent the dynamic of vibration mode(). Such a lumped model ha been developed by Figure.: Meaured (olid line) and predicted (dahed line) tranfer function of ball crew drive (ource: Varanai and Nayfeh [68]). 7

22 (a) (b) Figure.: Hyterei behaviour of the ball crew nut (ource: Cuttino et al. []): (a) Angular diplacement hyterei, (b) Friction torque hyterei. Varanai and Nayfeh [68] analytically, repreenting the t mode of torional vibration. The frequency repone function (FRF) they predicted with their model wa verified to be conitent with the meaured firt axial reonance, a hown in Figure.. Similarly, a lumped model wa developed to invetigate vibration compenation trategie for the firt axial mode in thi thei [3]. Although it i relatively traightforward to match ingle-mode lumped model to experimental data, higher order model require the ue of a modal analyi package to accurately identify the modal parameter and vibration magnitude contributed by each mode to the final repone of the drive...4 Motion Delivery in the Preloaded Nut The motion delivery in the preloaded nut exhibit a hyterei type nonlinear behavior originating from the rolling, lippage, and elatic deformation of the recirculating ball. Lin et al. [43] conducted a detailed kinematic analyi revealing that the teel ball continuouly undergo micro-cale lip between the crew and the nut during the motion tranmiion. Cuttino et al. [] purued thi work and uccefully modeled the elatic deformation of the ball, related lip phenomena, and hyterei behavior in the nut mechanim. Thi motion lo can be oberved in the nut linear poition v. ball crew angle and alo in the friction torque v. ball crew angle profile, a een in Figure. []. The nonlinear motion lo in the nut produce a tranlational poition error if the control loop i olely cloed with angular encoder. Thi effect can be alleviated to a certain extent by contructing compenation function or uing a linear encoder for direct tranlational feedback. Both approache have been followed in thi thei. Since thi topic wa earlier tudied in detail by other reearcher, elaborate modeling of the nut dynamic ha been kept outide the cope of thi thei. 8

23 . Control Law Deign Thi ection review ome of the contribution on feed drive control. Simpler technique like PID or P-PI velocity cacade control are typically implemented by either trial and error tuning, or uing imple plant model that capture only rigid body dynamic. More advanced technique, on the other hand, might require modeling of the axial and torional vibration, hyterei behavior in the nut, nonlinear friction characteritic, lead error, and motor torque ripple. In return, more detailed model can improve the control law deign and ultimately the tracking performance, by helping compenate for uch effect. Another important factor in the controller deign i robutne againt external diturbance (i.e. cutting force) and change in the dynamic characteritic of the drive (e.g. vicou friction, inertia, etc.)... Tracking veru Contouring Control Motion control ytem fall into two major categorie, which are tracking and contouring control. In multi-axi machine, tracking control refer to independent poition control of each axi drive uing it own feedback. Contouring control, on the other hand, i the cae where the contour error (i.e. geometric deviation from the deired toolpath) i directly etimated and ued in the feedback law. Koren [36] initiated the work in contouring control by developing the Cro-Coupling Controller (CCC). Thi initial deign conidered olely linear toolpath in etimating the contour error. Thi algorithm wa later extended to accommodate circular and parabolic toolpath in [37]. Erkorkmaz and Altinta [4][5] preented a real-time contour error etimation method for arbitrarily haped toolpath, and implemented it uccefully in a CCC cheme in conjunction with Zero Phae Error Tracking Control (ZPETC) [65]). Thi helped to imultaneouly reduce the contouring and tracking error. Following the concept of directly reducing the contouring error rather than individual axi tracking error, Chiu and Tomizuka [9] devied a cheme in which the multi-axi dynamic of the machine tool i projected along the tangential and normal direction of the toolpath. They deigned a controller to reduce the tracking error in the normal and bi-normal direction of motion, which in turn helped to reduce the contour error. In thi thei, the control algorithm are developed following the tracking control chool of thought. Thi i a more practical approach compared to applying contouring control, epecially conidering that the robotic and machine tool indutrie are currently moving toward reconfigurable deign which require more flexibility and interchangeability in the control ytem implementation. The objective i to indirectly minimize the contour error, by minimizing the tracking error in the individual axe. 9

24 .. Controller Deign for High Accuracy and Vibration Avoidance The mot favorable characteritic in a ervo ytem are the uppreion of diturbance and parameter variation effect for a wide frequency range, robutne againt unmodeled high frequency dynamic and meaurement noie, and a wide command following bandwidth. Pritchow noted the importance of achieving high cloed-loop bandwidth in order to track udden change in the reference trajectory and alo to reject diturbance caued by cutting (i.e. work) force and friction in [53]. Although many controller have been developed baed on rigid body dynamic alone, neglecting the influence of tructural reonance ignificantly limit the achievable bandwidth [54]. For example, the flexibility of a ball crew caue torional and longitudinal (i.e. axial) vibration which can be excited through the control ignal or diturbance caued by the cutting force or friction. If the dynamic that govern thee vibration are not conidered in the control law deign, intability occur when the feedback gain are increaed too much for tight poition tracking. One olution that alleviate thi problem to a certain extent i to inert a notch filter into the control loop. Although the notch filter help to attenuate the reonance effect, it doe little to recover the phae margin, which till ultimately limit the achievable bandwidth. Furthermore, if the reonance frequency hift due to a change in the ma of tiffne parameter, the cloed-loop performance or even tability may be lot. Neverthele, a ignificant improvement can be obtained by uing a notch filter, compared to the pure rigid body-baed deign cae. In machine tool drive, Smith [6] ha uccefully implemented notch filtering a a olution to thi problem. It i important to note that even with notch filtering, tructural vibration are not completely eliminated due to the exitence of external diturbance force and uncertaintie in the drive model. On the other hand, active vibration damping, if implemented correctly, can reult better ervo performance by attenuating the tructural vibration through the ue of feedback. Another preventative meaure in avoiding tructural vibration i Input Command Pre-haping (ICP) (or Input Shaping). Early invetigation of thi approach wa conducted by Smith [6] through the ue of Poicat control (Smith imagined the correponding control motion a cating a fly, and hence named it Poitive-cat or Poicat). In input haping, the input command i modified (i.e. prehaped) through the contruction of a erie of impule in order to achieve vibration-free poitioning [9], uing a dynamic model of the ytem. The method i baed on the idea that uperimpoed impule repone will be cancelled out by each other after the lat input impule i applied [7]. Hyde and Seering [5] preented the theory behind ICP. For a imple cae of a equence with two impule, they howed that the equence conit of an impule and a econd impule with a leer magnitude occurring after half the period of the damped vibration. The action of adding an

25 appropriately delayed impule repone, which i performed by the ICP method, i equivalent to adding zero at the ytem mode. Similarly, Jone and Uloy [9] viewed the ICP technique a a pole-zero cancellation problem and found that it i equivalent to a pecial cae of feedforward notch filtering of the poition command. Feedforward control can alo be ued to cancel out the table dynamic of a cloed-loop ytem, thereby yielding an overall command following tranfer function that i very cloe to unity for a wide frequency range. One of the notable contribution in thi area came from Tomizuka [65], who introduced the Zero Phae Error Tracking Controller (ZPETC) which i baed on cancellation of all of the cloed loop pole and table cloed loop zero. Theoretically, thi controller yield zero phae hift between the commanded and actual poition value, and a gain that i very cloe to unity for a wide frequency range. A imilar contribution wa alo made by Weck and Ye [69]. However, the performance of feedforward controller i very enitive to the accuracy of the aumed dynamic model [5], a ha been hown by Pritchow and Philipp. In reality, a drive dynamic, even if they can be identified in detail [66], can vary over time and cannot be kept unchanged. The mot important hortcoming of any type of feedforward approach i it open loop nature, which require cloed-loop robutne to be realized in feedback before it can be uccefully applied. A mentioned earlier, a more effective way to deal with tructural vibration i to attempt to attenuate them in feedback control. Thi facilitate the achievement of a wider cloed-loop bandwidth and better diturbance rejection. Chen and Tluty were among the firt to tudy thi concept for machine tool drive, in which they demontrated the effectivene of vibration damping uing accelerometer feedback in imulation [8]. In recent year, a more powerful control computer and high reolution feedback device have became available, ucceful experimental reult have been reported by reearcher for ball crew drive. Symen et al. [64] ued H robut control with gain cheduling, which helped track the variation in tructural dynamic with axi poition. Zatarain et al. [7] improved the drive tiffne, damping, and tool poitioning accuracy by fuing linear encoder meaurement with accelerometer feedback through a Kalman filter. Accelerometer feedback wa alo ued by Symen et al. A a contribution from indutry (Siemen AG), Schäfer [56] developed a control loop tuning trategy which emulate the behavior of a mechanical damper, allowing high jerk movement to be executed without vibrating the tranlational part of the drive. All of thee method enable a ignificant improvement in the command following bandwidth, dynamic (i.e. frequency dependent) tiffne, and diturbance rejection characteritic. Recently, Kamalzadeh and Erkorkmaz have developed an active vibration damping and poition control trategy for ball crew drive uing Adaptive Sliding Mode Control (ASMC) [3]. Thi

26 approach utilize the methodology et forth by Slotine and Li [6] and actively compenate for the dynamic of the t axial mode. One nice feature about thi controller i that it inherently contain the command following (feedforward) and diturbance rejection (robutne) term that are yntheized through the olution of one Lyapunov inequality. Compared to other approache, thi control law i relatively eay to deign and ha been demontrated to achieve a high command following bandwidth (8 Hz), and good accuracy (.3 um) during high travel peed and acceleration ( mm/ec,.5 g). Thi controller deign i one of the main contribution in thi thei, and i preented in Chapter 6. ASMC ha alo been ued to control the rigid body dynamic in thi thei. In thi cae, the vibratory mode have been attenuated by inerting notch filter into the control loop. Active vibration cancellation for the t torional mode, through additional control term, ha alo been invetigated and reported in []. The idea of Sliding Mode Control wa pioneered by Utkin [67]. The original idea wa baed on dicontinuou witching of the control ignal in order to keep the ytem tate on a liding urface, which repreent a table linear differential equation governing how the tate hould converge to the origin (or to their deired value). Hence, the cloed-loop repone i dictated by the parameter of the liding urface. Slotine and Li [6] refined thi methodology by developing a general and adaptive framework, which reulted in a continuou and nonlinear control law with Lyapunov-guaranteed tability. Later Zhu et al. [75] provided the tability proof for conducting the parameter adaption within known bound. In the early tage of the reearch in thi thei, many control technique were invetigated uch a Pole-Placement, ZPETC, and ASMC. Adaptive Sliding Mode Control wa choen a the foundation for everal of the developed compenation technique, due to it excellent tracking performance and robutne againt diturbance and low frequency dynamic variation...3 Compenation of Torque/Force Ripple Variation of the motor torque due to change in the relative poition between the rotor and tator i known a the motor torque ripple. Diturbance, uch a motor and mechanical torque ripple, caue inaccurate torque delivery from the drive which in turn can deteriorate the motion control accuracy. Torque ripple can be modeled and compenated a a function of the motor poition and the torque command [3][5]. In AC ervo-drive, which are becoming the maintream in machine tool and indutrial robotic application, the torque ripple i compoed of cogging torque, reluctance torque, mutual torque, and

27 the DC current offet torque [4]. Cogging torque originate from the preence of tator lot and ha a frequency at which the lot are located along the tator []. Reluctance torque i due to the rotor geometry dicontinuity (hole, aliencie, and other geometrical dicontinuitie which interrupt the internal flux). Thi caue variation in reluctance a the rotor rotate. Mutual torque i due to the interaction between the rotor magnetic field and tator current. Ideally, a motor hould produce a contant mutual torque if the tator winding and rotor magnetic field have a perfect inuoidal ditribution. However, ince thi i not achievable in practice, the produced mutual torque contain higher harmonic which caue the torque ripple [5]. The torque ripple due to DC current offet i caued by the offet in the current enor and digital to analog converter (DAC) in the current control loop. In optimally deigned AC permanent magnet motor, cogging, reluctance, and mutual torque ripple can be neglected [3]. The DC current offet ripple are more dominant among other mentioned factor, due to the offet in the current enor and DAC being difficult to eliminate [4]. Mechanical torque ripple, on the other hand, can be oberved when there are minor mialignment in the feed drive mechanim. Many technique have been propoed in literature for torque ripple minimization. The firt cla of thee technique focue on improving the motor deign [7][8][39]. The econd cla conider injecting additional control ignal for canceling out the ripple, by upplying an equivalent current to the motor. Thi cancellation can be performed in the current or motion control loop. A an example of torque ripple compenation in the current control loop, Parailiti et al. [5] ued a Kalman filter [3] to identify the harmonic component of the magnetic flux linkage and predict the neceary motor current to compenate for the ripple. However, to implement uch an algorithm, the current amplifier hardware need to be modified. In thi thei, torque ripple compenation i realized at the poition and velocity control level, which i one level higher and more acceible. Thi work ha been publihed in [3]. One of the advantage of thi approach i that it allow the methodology to be applied on different type of feed drive in a generic manner, without having to modify the current loop. The torque ripple harmonic were identified uing a Kalman filter diturbance oberver. Detail of thi work are preented in Chapter Compenation of Elatic Deformation Elatic deformation occur in ball crew drive typically due to inertial force, guideway friction, and cutting force, which reult in elongation and compreion of the ball crew. Many of the ball crew drive are controlled baed on cloing the poition loop only with a rotary encoder, a thee encoder are le expenive compared to linear encoder. When only rotary poition feedback i ued, the linear 3

28 poitioning accuracy uffer due to lead error, elatic deformation, thermal deformation, and motion lo in the preloaded nut [][3][6][59][7]. By uing direct poition feedback through a linear cale, it i poible to alleviate thee problem to a certain extent, mainly during teady-tate poitioning and low frequency movement. However, during high peed and acceleration, thee factor become apparent again; and can deteriorate the dynamic linear accuracy of the drive mechanim. Alo, with the ue of linear feedback, the incluion of additional mechanical flexibility (i.e. the axial mode) inide the ervo loop can caue the bandwidth to uffer [8][54][68] if the tructural reonance are not compenated adequately. Although technique have been developed to avoid or attenuate tructural vibration by pre-filtering the poition command or inerting notch filter inide the loop [][7][9][6][69], thee technique uually reult in a drop in the poition tracking bandwidth compared to uing collocated rotary feedback. Moreover, linear encoder typically cot ten time more compared to rotary encoder, and their intallation involve additional expene a well. Among the above mentioned factor, compenation of the elatic deformation ha received attention in motion control literature. In the robotic field, Zhang et al. tudied the joint torque feedback technique [73] uing a torque enor in a flexible robot arm mechanim. The joint torque refer to the joint diplacement multiplied by the tiffne of the flexible tranmiion ytem. Thi i conceptually equivalent to the tranmitted torque that i etimated in the elatic deformation compenation cheme preented in Chapter 7 of thi thei. In [73], the author invetigated the idea of injecting negative joint torque feedback to the control ignal, and the relation between vibration uppreion and diturbance rejection, which refer to the compenation of elatic deformation caued by diturbance. While a negative joint torque feedback i effective in vibration uppreion (conitent with Chen and Tluty imulation [8] for negative acceleration feedback), it amplifie the effect of diturbance on the end effector poition [74]. On the other hand, poitive joint torque feedback improve the diturbance rejection characteritic, but ha a detabilizing effect on the overall ervo ytem. A imilar effect wa alo oberved in the feedback baed elatic deformation compenation cheme developed in Chapter 7. However, in thi thei, the intability problem ha been mitigated by deigning additional filter and applying a detailed tability analyi. Lim et al. [4][4] adopted the idea of poitive joint torque feedback from Zhang et al. [74], and applied it on an X-Y poitioning table. Their method work in a feedback manner and reject the diturbance that caue elatic deformation. However, the performance of thi method i limited due to two major hortcoming. Firt, thi technique aume a contant tiffne parameter for the ball crew drive and doe not take into account it poition dependent flexibility. Second, the effect of 4

29 vicou and Coulomb friction are neglected in computing the tranmitted (i.e. joint) torque, which can lead to incorrect etimation of the elatic deformation. In thi thei, more comprehenive and yet practical elatic deformation compenation trategie have been developed that addre the hortcoming of earlier work. A feedforward approach ha been developed [33], which conider both the poition dependency and the ource of external friction in computing the elatic deformation. Thi approach i relatively imple but not very robut againt parameter change in the load ide of the drive, uch a table ma and guideway friction variation. A feedback baed technique ha alo been developed [34], which i more robut and till relatively eay to implement. Thi technique i conidered to be another major contribution in thi thei. Both approache are detailed in Chapter 7, which deal with the compenation of elatic deformation in ball crew drive..3 Concluion Thi chapter ha preented a urvey of ome of the critical iue related to the modeling and control of ball crew drive. Namely they relate to the rigid body dynamic, friction, elatic deformation, torque ripple, lead error, thermal deformation, tructural vibration, and motion lo in the preloaded ball-nut mechanim. In the following chapter, ome of thee iue will be tudied in more detail. New control law will be developed which focu on achieving high command following and diturbance rejection bandwidth through the avoidance or damping of tructural vibration. Alo, the achievement of high poitioning accuracy will be invetigated by compenating for ome of the repeatable effect uch a torque ripple, lead error, and elatic deformation. 5

30 Chapter 3 Modeling and Identification 3. Introduction Development of high performance motion controller require a good undertanding of the feed drive dynamic. In thi chapter, the mot ignificant dynamic of a ball crew drive, which i ued a the primary experimental etup in thi thei, are modeled and experimentally identified. Section 3. detail the deign of the experimental etup. Section 3.3 focue on the identification of rigid body dynamic, including effect like inertia, vicou friction, and Coulomb friction. The drive parameter, including Coulomb friction torque, are initially calculated uing a Leat Square technique. The friction model i then refined by oberving the control ignal equivalent friction diturbance uing a Kalman filter, while jogging the drive under feedback control at different velocitie. The amplifier current loop i alo meaured, to validate the frequency range in which effective actuation can be realized. Section 3.4 deal with the modeling and identification of the vibratory dynamic. Torional vibration are invetigated uing finite element analyi and frequency repone teting. The firt axial vibration mode, which i highly ignificant in term of influencing the drive linear poitioning accuracy, i etimated through an analytical calculation and validated experimentally. The knowledge of torional and axial vibration mode i crucial in deigning high bandwidth motion controller. Concluion of thi chapter are preented in Section Experimental Setup The high peed ball crew drive built at the Preciion Control Laboratory, Univerity of Waterloo i hown in Figure 3.. The ball crew, which ha mm diameter and mm pitch, i ued to drive a table upported on an air guideway ytem. Actuation i provided through a 3 kw AC ervomotor which i connected to the ball crew uing a diaphragm-type non-backlah coupling. The motor i operated in current control mode. The bandwidth of the current control loop wa meaured to be 48 Hz. The ball crew i intrumented at both end with two high reolution rotary encoder which deliver 5 inuoidal ignal per revolution. Thee encoder ignal can be reliably interpolated by 4 time in the motion controller, reulting in a poition meaurement reolution equivalent to nm of table motion. Catalogue rated accuracy of thee encoder are equivalent to nm of table motion. For ucceful vibration meaurement, thee encoder have been mounted rigidly onto the ball crew and to the machine bae. Encoder i located at the free end of the ball crew, and Encoder i right 6

31 Figure 3.: Experimental etup. after the fixed bearing upport, a een in Figure 3.. A third encoder (Encoder 3) i available on the back of the motor which deliver 6384 quadrature pule, reulting in a poition meaurement reolution equivalent to 35 nm of table motion. Thi tudy wa initially focued on rotary dynamic of the drive, uing the meaurement obtained from the rotary encoder. Later on, to invetigate the axial vibration and validate the final linear poitioning accuracy of the table, a linear encoder with 4 um ignal period wa retrofitted on the table which provide a meaurement reolution of nm and an accuracy of 4 nm, a rated in the catalogue. The high reolution of the encoder enable accurate detection of the frequency repone at different point along the ball crew, by yielding clear acceleration ignal after double differentiation with repect to time. Hence, the vibration mode hape can be oberved and verified up to a frequency of.5 khz. The etup i controlled with a dspace controller which can achieve a ampling frequency up to khz. 3.3 Rigid Body Dynamic Identification The rigid body dynamic comprie the mot fundamental characteritic of ball crew drive, which dominate the low frequency behavior. Accurate knowledge of thee dynamic i eential for any kind of model-baed controller deign. Furthermore, thee characteritic hould be determined before 7