Precision Control of High Speed. Drives using Active Vibration Damping

Transcription

1 Precision Control of High Speed Drives using Active Vibration Damping b Daniel Gordon A thesis presented to the Universit of Waterloo in fulfillment of the thesis requirement for the degree of Master of Applied Science in Mechanical Engineering Waterloo, Ontario, Canada, Daniel Gordon

2 AUTHOR'S DECLARATION I hereb declare that I am the sole author of this thesis. This is a true cop of the thesis, including an required final revisions, as accepted b m examiners. I understand that m thesis ma be made electronicall available to the public. ii

3 Abstract In order to meet industr demands for improved productivit and part qualit, machine tools must be equipped with faster and more accurate feed drives. Over the past two decades, research has focused on the development of new control strategies and smooth trajector generation techniques. These developments, along with advances in actuator and sensor technolog, have greatl improved the accurac of motion deliver in high speed machine tools. However, further advancement is limited b the vibration of the machine s structure. The purpose of the research in this thesis is to develop new control techniques that use active vibration damping to achieve bandwidths near the structural frequencies of machine tools, in order to provide better dnamic positioning of the tool and workpiece. Two machine tool drives have been considered in this stud. The first is a precision ball screw drive, for which a pole-placement technique is developed to achieve active vibration damping, as well as high bandwidth disturbance rejection and positioning. The pole-placement approach is simple and effective, with an intuitive phsical interpretation, which makes the tuning process straightforward in comparison to existing controllers which activel compensate for structural vibrations. The tracking performance of the drive is improved through feedforward control using inverted plant dnamics and a novel trajector prefilter. The pre-filter is designed to remove tracking error artifacts correlated to the velocit, acceleration, jerk and snap (4 th derivative) of the commanded trajector. B appling the least-squares method to the results of a single tracking experiment, the pre-filter can be tuned quickl and reliabl. The proposed controller has been compared to a controller used commonl in industr (P-PI position-velocit cascade control), and has achieved a 4-55 percent reduction in peak errors during tracking and machining tests. The controller design, stabilit analsis, and experimental results are discussed. The second drive considered is a linear motor driven X-Y stage arranged as a T-tpe gantr and worktable. The worktable motion is controlled independentl of the gantr using a loop shaping filter. The gantr is actuated b dual direct drive linear motors and is strongl coupled to the worktable position, which determines its inertial characteristics. A 94 Hz aw mode is handled in the gantr control law using sensor and actuator averaging, and active vibration damping. The stabilit and robustness of the design are considered using multivariable frequenc domain techniques. For the worktable motion along the gantr, a bandwidth of 3 Hz is achieved. The gantr crossover frequenc is 5 Hz, which is 3 times higher than the bandwidth that can be achieved using independent PID controllers (6 Hz). The iii

4 performance of the proposed control scheme has been verified in step disturbance (i.e., rope snap) tests, as well as tracking and contouring experiments. iv

5 Acknowledgements I would like to express m gratitude to m supervisor, Dr. Kaan Erkorkmaz, for the guidance and knowledge he has provided throughout m studies. His enthusiasm for his work is inspiring, and it has been a pleasure working with him. I would also like to thank m friends in the Precision Controls Laborator, as well as the facult and staff of the Mechanical and Mechatronics Engineering department. In particular, I would like to thank Jeff Gorniak, Dana Chan and Yasin Hosseinkhani for man helpful conversations. Also, thanks to Amin Kamalzadeh, for his help in the earl stages with the experimental setup, and to AJung Moon and Sui Gao for their help in preparing the ball screw for the machining experiments. I would also like to thank Robert Wagner, Jason Benninger and Neil Griffett for their help with the experimental setups. Finall, I would like to gratefull acknowledge the support of m famil in making m graduate studies possible. v

6 Dedication To Monica and m parents vi

7 Table of Contents AUTHOR'S DECLARATION... ii Abstract... iii Acknowledgements... v Dedication... vi Table of Contents... vii List of Figures... ix List of Tables... xii Chapter Introduction... Chapter Literature Review Modeling and Identification Rigid Bod Dnamics Structural Vibrations Friction Control Law Design and Active Vibration Damping Feedforward Control Conclusions... 7 Chapter 3 Control of Ball Screw Drives Introduction Controller Design Drive Model Pole-Placement Feedback Design State Trajector Generation and Model Inversion... 6 vii

8 3..4 Trajector Pre-filtering Frequenc Domain Analses Experimental Results Conclusions... 9 Chapter 4 Control of a Linear Motor Driven Gantr Introduction Experimental Setup Modeling and Identification X-Axis Modeling and Identification Y-Axis Modeling Y-Axis Identification Controller Design X-Axis Controller Design Y-Axis Controller Design Experimental Results Conclusions Chapter 5 Conclusions Conclusions Future Research References viii

9 List of Figures Figure 3.: Proposed control scheme which compensates the st axial mode using pole-placement Figure 3.: Ball screw drive setup.... Figure 3.3: Simplified flexible drive model.... Figure 3.4: Measured and modeled drive FRF s with (a) excitation from motor and (b) excitation from impact hammer... Figure 3.5: Variation of sstem FRF with table position Figure 3.6: Open- and closed-loop pole locations Figure 3.7: Tracking error profile with velocit, acceleration, jerk and snap artifacts Figure 3.8: Block diagram manipulation for assessing closed-loop stabilit.... Figure 3.9: P-PI position-velocit cascade control scheme with velocit and acceleration feedforward terms.... Figure 3.: Loop magnitude of (a) MC-PPC and (b) P-PI; Nquist plot for (c) MC-PPC and (d) P-PI... 3 Figure 3.: Sensitivit magnitude for (a) MC-PPC and (b) P-PI Figure 3.: Disturbance transfer functions for MC-PPC (left) and P-PI (right) considering ideal plant.. 5 Figure 3.3: Command tracking with MC-PPC (left) and P-PI (right) with experimentall measured plant... 6 Figure 3.4: (a) Table displacement and (b) amplitude spectral densit of table displacement for the MC- PPC and P-PI in Rope Snap Test... 7 Figure 3.5: Tracking error, control signal, and FFT of tracking error for (a) mm/s, (b) 5 mm/s, (c) mm/s and (d) 5.4 mm slotting at mm/s in 665 aluminum Figure 4.: Linear motor driven T-tpe gantr Figure 4.: Schematic of the experimental setup Figure 4.3: Open-loop acceleration FRF for x-axis ix

10 Figure 4.4: Y-axis dnamic model Figure 4.5: Mass distribution variation with x-axis location Figure 4.6: Y-axis acceleration response Figure 4.7: Model identified for -axis position FRF with table at x = -.5 m (left), x =. m (center) and x = +.5 m (right) Figure 4.8: Impact hammer test results Figure 4.9: X-axis loop gain (left), sensitivit (center) and Nquist plot (right)... 4 Figure 4.: Y-Axis (gantr) control scheme Figure 4.: Contribution of active damping demonstrated in frequenc and time domain results. Tracking results are for a mm/s velocit, m/s acceleration, 5 m/s 3 jerk trajector Figure 4.: Maximum singular values of sensitivit function for various integral action gains (k i ) Figure 4.3: Y-axis loop transfer function upper and lower singular values (left) and MIMO Nquist plot (right) Figure 4.4: Upper and lower singular values of the sensitivit function for the proposed MIMO controller (left) and the SISO PID controllers (right) Figure 4.5: Singular values of open (G) and closed-loop (SG) transfer functions for the proposed MIMO controller (left) and independent SISO PID controllers (right) Figure 4.6: Experimental setup for step disturbance (i.e., rope-snap) testing... 5 Figure 4.7: Table -axis displacement, control signal and amplitude spectral densit for MIMO (proposed) and SISO PID controllers during rope-snap tests Figure 4.8: Y-Axis tracking performance for mm/s and mm/s for the proposed MIMO and SISO PID controllers Figure 4.9: X and Y-axis tracking performance for the proposed controller Figure 4.: Contouring results for. um feature sizes at um/s feedrate,.6 um/s acceleration and um/s 3 jerk x

11 Figure 4.: Contouring results for. mm feature sizes with mm/s feedrate,.6 mm/s acceleration and mm/s 3 jerk Figure 4.: Contouring results for mm feature sizes with mm/s feedrate, m/s acceleration, 5 m/s 3 jerk xi

12 List of Tables Table 3.: Identified parameters... Table 3.: Maximum and RMS errors during tracking and machining experiments... 9 xii

13 Chapter Introduction The manufacturing industr is continuall pushing for reductions in ccle times and more stringent tolerances, in order to maximize productivit and produce higher qualit products. As a result, the demands on machine tools continue to rise. In order to take advantage of improvements in other areas of machine tool engineering, such as increasing spindle speeds and the introduction of more advanced tooling, feed drives must be able to deliver higher speeds while maintaining or improving positioning accurac. Over the past two decades, research into new control strategies and smooth trajector generation, along with improvements in actuator and sensor technolog, have greatl enhanced the motion deliver provided b high speed feed drives. In order to realize the most favorable command tracking and disturbance rejection properties in feed drives, a high closed-loop bandwidth must be obtained [9]. Structural vibrations of the drive and machine tool can severel limit the control bandwidth. Some of the methods that have been proposed in literature to deal with such vibrations include pre-filtering of the motion commands [7] and notch filtering of the control signal to attenuate actuation near the resonances [33]. More recentl, controllers that use ball screw drives or linear motors to activel compensate for structural dnamics have been developed, as reported in [][9][3][34]. This approach enables higher bandwidths to be achieved, but the design becomes more difficult as higher order plant models must be considered. In ball screw drives, the first axial mode is the limiting factor in achieving high control bandwidths. In this thesis, a new method of providing active vibration damping for ball screw drives using poleplacement control is introduced. This method uses a phsicall meaningful model and provides an intuitive choice of closed-loop pole locations. On the other hand, machines which utilize direct drive linear motors are increasing in popularit. Direct drives have man advantages over conventional ball screw drives in terms of dnamic response and accurac. However, the are also more susceptible to disturbance forces and inertia variations, due to the lack of gear ratio between the motor and load. Some of these direct drive machines are arranged in gantr configurations, which result in one axis being actuated b parallel drives. These drives will inevitabl be coupled through the gantr inertia and vibrations. As a result, achieving a high control bandwidth and ensuring stabilit is a significant challenge.

14 In this thesis, a dnamic model for a T-tpe gantr has been developed, which captures the dnamics of the first vibration mode. Using insight from this model, a multi-input, multi-output (MIMO) control law consisting of a sensor/actuator averaging scheme with active vibration damping has been designed. The subsequent chapters in this thesis are arranged as follows: A literature review on existing modeling, identification and control techniques for machine tool feed drives is presented in Chapter. In particular, modeling and control of higher order dnamics is relevant to the goal of this thesis. Modeling of rigid-bod dnamics and nonlinear friction, as well as feedforward controller design have also been discussed. In Chapter 3, pole-placement control is used to activel dampen the st mode of axial vibrations in a ball screw drive. Trajector tracking is improved through feedforward control, using inverted plant dnamics, feedforward friction compensation, and a Least Squares tuned trajector pre-filter. Effectiveness of the proposed strateg is verified in frequenc domain analses and also in rope-snap testing, trajector tracking and metal cutting experiments. The proposed scheme is compared to the mainstream P-PI cascade design used in industr. The sensor and actuator averaging concept has been adopted in a novel controller design for a T-tpe gantr in Chapter 4. The control law has been designed to take into account the dnamic coupling between the actuators and to activel attenuate the first vibration mode encountered at 94 Hz on the gantr. The stabilit and robustness of the design is achieved through the use of multivariable frequenc domain analses.

15 Chapter Literature Review This chapter is dedicated to reviewing some of the work done b other researchers which is relevant to the topics addressed in this thesis. Section. reviews past research into the modeling and identification of ball screw drives and direct-driven machine tools. Different approaches to the feedback controller design of these machines have been presented in Section.. Finall, Section.3 covers feedforward control concepts.. Modeling and Identification Much research has been devoted to the modeling of ball screw drives and direct-driven machine tools, including the modeling of their structural flexibilit. Dnamic modeling can be performed at man different levels of detail and complexit, ranging from rigid bod models to complicated models built using finite element methods (FEM). In this section, modeling of the rigid bod dnamics, structural vibrations, and nonlinear friction are discussed... Rigid Bod Dnamics Rigid bod modeling attempts to capture the low frequenc behavior of the sstem, basicall consisting of the effects of inertia, viscous damping and Coulomb friction effects. Direct drives are generall not prone to the low-frequenc modes that plague ball screw drives. This allows them to generall be modeled as a pure mass with viscous damping [37]. This is one of the approaches emploed in Chapter 4. However, when direct drives are arranged in gantr configurations, crosstalk through the inertia and low-frequenc structural vibrations of the gantr is inevitable. In literature, Teo et al. [35] have developed a rigid bod dnamic model for an H-tpe gantr, neglecting vibration modes and stiffness values... Structural Vibrations Structural vibrations in machine tools can severel limit the controller performance. In particular, if no method of avoiding excitation or damping of the vibrations is implemented, the achievable closed-loop bandwidth will be significantl below the frequenc of the st vibration mode [8]. In order to appl 3

16 vibration compensation, knowledge of the vibration mode is essential. In literature, approaches to modeling the structural vibrations include analtical methods [6][38], frequenc response measurement [8][33][38] and finite element methods [][8][6][33][4]. A lumped model can be used to represent the vibrator dnamics, such as the one developed analticall b Varanasi and Nafeh [38] for the st axial mode of a ball screw. A similar approach using a -DOF model with a diagonal mass matrix was used b Kamalzadeh and Erkorkmaz [9] to model the st axial mode of a ball screw. This model is used in Chapter 3 of this thesis. A -DOF model with a full mass matrix and least-squares identification using experimental frequenc response data similar to the method used b Okwudire in [7] is used in Chapter Friction Friction is a significant source of disturbance in machine tools. In particular, friction causes errors during motion reversals, since the controller integral action cannot immediatel respond to the discontinuous change in the friction force as the velocit crosses through zero. Accurate identification and feedforward compensation can help to reduce these positioning errors. Friction modeling and compensation techniques that include the Stribeck effect [4] have been presented in literature [4][9][]. The nonlinear friction disturbance, in simplified form, can be modeled as, d d( v) d c c ( d ( d s s d d c c ) e ) e v / v / when v when v when v (.) where d c is the Coulomb friction, significantl limits the achievable bandwidth [8]. 4 d s is the static friction, v is the velocit and is the velocit constant. A method of identifing the model parameters using Kalman filtering [8] to estimate the equivalent disturbance has been used in this work following the method in [9]. Feedforward friction compensation using the identified model is implemented in Chapter 3.. Control Law Design and Active Vibration Damping The main objectives of feedback control are to reject disturbances and accuratel follow commands. In [9], the importance of a high closed-loop bandwidth for command tracking and disturbance rejection was noted. Designing a controller based on rigid-bod dnamics and neglecting structural resonances

17 Notch filters in the control loop can also be used to mitigate the effect of structural vibrations. Using this method, Smith [33] has shown significant improvement over a pure rigid-bod based design. However, the notch filters have an adverse effect on the loop phase, and do not completel eliminate the structural vibrations due to external disturbances or modeling uncertainties. Conversel, active vibration damping can result in better performance b attenuating the structural resonances using feedback. Chen and Tlust proposed a method for active vibration damping using accelerometer feedback and showed its effectiveness in simulations [6]. As computer processors and feedback resolution have improved, successful experimental results have been reported b several researchers. Smens et al. [34] used H control with gain scheduling to account for the position dependenc of the structural vibrations. Zatarain et al. [43] combined linear encoder and accelerometer feedback using a Kalman filter to improve drive stiffness and damping. Another approach, presented b Schäfers [3], is to tune the proportionalintegral controlled velocit loop to emulate a vibration absorber. Kamalzadeh and Erkorkmaz [9] have developed an adaptive sliding mode controller (ASMC) for active vibration damping which activel compensates for the dnamics of the st axial mode of a ball screw drive. The disturbance adaptive discrete-time sliding mode controller (DADSC) proposed b Won and Hedrick [4] has been extended b Altintas and Okwudire to activel compensate for the axial mode in ball screw drives [5] and for damping of structural vibrations in direct feed drives []. Recentl, Gordon and Erkorkmaz have developed a method of activel damping the axial mode in ball screw drives using a pole-placement controller [5]. The most crucial step in pole-placement controller design is the choice of desired closed-loop pole locations. Some insight into desirable pole locations for suppressing the axial vibrations comes from the concept of low authorit control, as described in []. Low authorit control seeks to suppress resonance peaks while minimall affecting the transfer function awa from the resonance. In the root-locus, this corresponds to a change in the real part of the complex poles, without significantl changing the imaginar part. When the closed-loop pole locations have been chosen, the feedback gains can be calculated using Ackermann s formula [][4] or other more robust pole-placement algorithms []. Teo et al. [35] have reported successful experimental results on a ball screw driven gantr using adaptive control based on rigid bod dnamics. Weng et al. [4] have proposed the concept of sensor and actuator averaging for controlling flexible structures, in particular beams and plates. The sensor/actuator averaging acts as a spacial filter and can be used to attenuate resonances without the phase penalt associated with 5

18 notch filtering. This concept has been extended b Gordon and Erkorkmaz [4] to control a linear motor driven gantr, which is presented in Chapter 4..3 Feedforward Control Feedforward control can be used to enhance command tracking, as well as avoid excitation of structural resonances. However, feedforward control is sensitive to the accurac of the underling model and not robust against disturbances or changes in the plant [3]. Before appling feedforward control, closed-loop robustness should be realized through feedback control. It is common practice to make a feedforward controller approximate the inverse of the plant in order to minimize the transfer function from the reference to the error. However, model inversion is not alwas possible, notabl in sstems with non-minimum phase (unstable) zeros. Since man sstems are dominated b rigid-bod behavior, use of acceleration feedforward is widespread. Acceleration feedforward generates a control signal equal to the modeled mass times the commanded acceleration. For man sstems, appling this scheme leaves an error profile with correlated to the derivatives of acceleration. Boerlag et al. [5] have extended acceleration feedforward to the derivative of jerk (i.e., snap), in order to compensate for the residual servo errors. A new method of removing these artifacts through trajector pre-filtering was also developed b Gordon and Erkorkmaz in [5]. This approach is detailed in Chapter 3, and has been used in both Chapter 3 and Chapter 4 Tomizuka [36] developed the zero phase error tracking controller (ZPETC) to cancel out the poles and stable zeros of a closed-loop sstem, and approximatel cancel out the non-minimum phase zeros. This theoreticall results in a zero phase lag and unit gain for a wide frequenc range. Others, including Weck and Ye [39], have also developed similar techniques. These methods result in non-causal sstems. Feedforward control can be used to avoid exciting structural vibrations. One such technique that has been demonstrated is input command pre-shaping (ICP) or input shaping. Based on a dnamic model of the sstem, ICP generates a series of impulses such that the oscillator parts of the impulse response cancel out, allowing vibration free positioning to be achieved [6]. Jones and Ulso [7] found that ICP was a special case of notch filtering the position commands, and have successfull applied this strateg o controlling coordinate measuring machines (CMM s). 6

19 .4 Conclusions This chapter has presented a review of the work done previousl in areas related to the thesis topic. The various approaches to modeling and control of machine tool feed drives have been presented. In general, the approach used in this thesis has been to make use of the simplest tools available which still provide effective results. 7

20 Chapter 3 Control of Ball Screw Drives 3. Introduction In this chapter, pole-placement control is used to activel dampen the st mode of axial vibrations in a ball screw drive. Trajector tracking is improved through feedforward control, using inverted plant dnamics, feedforward friction compensation, and a Least Squares tuned trajector pre-filter. Effectiveness of the proposed strateg is verified in frequenc domain analses and also in rope-snap testing, and trajector tracking and metal cutting experiments. The proposed scheme is compared to the mainstream P-PI cascade design used in industr. 3. Controller Design The proposed overall control scheme is shown in Figure 3.. The feedback loop is closed using poleplacement control, which provides active vibration damping for the axial mode, as well as motion control. Feedforward terms, including inverted sstem dnamics and Stribeck-tpe friction compensation are included in order to improve command tracking. A new trajector pre-filter has been developed, which removes the artifacts of the commanded velocit, acceleration, jerk and snap from the tracking error. Finall, a filter pack comprising of notch and low-pass filters is added to ensure stabilit. In the following subsections, the purpose and design of the individual components will be explained. Their individual contributions will be validated through frequenc domain analses in Section 3.3 and shown in experimental results in Section

21 Figure 3.: Proposed control scheme which compensates the st axial mode using pole-placement. 3.. Drive Model The ball screw drive test bed is shown in Figure 3.. The table is supported b an air guidewa sstem and driven b an NSK WFA-3P-C5Z precision ball screw with mm pitch and mm diameter. A 3 kw AC servomotor provides the actuation through a diaphragm tpe coupling. Feedback is provided b a high resolution rotar encoder which is mounted on the ball screw and a linear encoder mounted on the table. The rotar encoder produces 5 sinusoidal signals per revolution which can be interpolated b 4 times, giving a position measurement resolution equivalent to nm of table motion. The catalogue rated accurac of this encoder is equivalent to nm of table motion. The linear encoder, which has a rated accurac of 4 nm, has 4 um signal period and is also interpolated 4 times, providing a measurement resolution of nm. The control laws are implemented at khz sampling frequenc on a dspace sstem. 9

22 Figure 3.: Ball screw drive setup. Figure 3.3: Simplified flexible drive model. The controller is designed considering the simplified dnamic model shown in Figure 3.3, which is captured b Equation (3.): m x m x b x k b x x x cx x u kx x cx x d As shown in earlier work [9], this model captures the dnamics of the st axial mode with sufficient closeness. If necessar, more accurate models can also be used with the proposed PPC scheme, such as those presented in [5][38] which have a full mass matrix and take the form, d (3.)

23 m m m m c x c c c k x k k x F k (3.) In Figure 3.3, the inertias of the rotating and translating components are denoted b m and m. k represents the overall axial stiffness, which is influenced b the nut, the fixed bearing, and torsional and axial stiffness of the screw. The damping in the preloaded nut is represented b c. Viscous damping at the motor and screw bearings, and table guidewas, is captured with b and b. The motor torque is represented in terms of the equivalent control signal u [V]. d and d correspond to equivalent disturbances acting on the rotating and translating components. The position units are radians, which represent equivalent rotational motion of the screw. The identified control signal equivalent parameters are shown in Table 3.. Table 3.: Identified parameters Parameter Identified Value Units m V/(rad/s ) m V/(rad/s ). V/(rad/s ) b 3 b V/(rad/s ) k. V/rad c. 4 V/(rad/s ) This model was validated b comparing the acceleration FRF measured through the rotar and linear encoders with the model FRF, as shown in Figure 3.4a. As can be seen, the st axial mode at 3 Hz is adequatel captured. However, the gain and phase agreement deteriorates after ~5 Hz due to additional dnamics contributed b the torsional modes (44, Hz), and the bandwidth of the drive s current control loop, which was identified to be 45 Hz. The identified drive model comprises of complex conjugate poles s j ) at 3 Hz with.7% damping, a real pole at.7 Hz due to the (, d interaction of viscous friction with the total inertia, and an integrator for velocit to position conversion.

24 Figure 3.4: Measured and modeled drive FRF s with (a) excitation from motor and (b) excitation from impact hammer. The disturbance response of the drive was measured through impact hammer testing and is shown compared to the model response in Figure 3.4b. A small resonance at 43 Hz is noted in the disturbance FRF. Further testing revealed that this mode corresponded to the pitching motion of the table on the clindrical guidewas. The ball screw was mounted at the same level as the guidewas; hence this mode cannot be excited b the motor and therefore it is not considered in the controller design. However, it does appear in the disturbance response observed in the experimental results. The discrepanc between the modeled and measured disturbance response is postulated to be due to the lack of excitation provided b the small instrumented hammer used for this measurement. Further investigation with a larger hammer is planned. To gauge the position dependenc of the ball screw s response, FRF measurements were taken, with the excitation from the motor, spanning most of the travel range of the ball screw at, 5 and 3 mm. As seen in Figure 3.5, the frequenc response of the drive does not var significantl with table position. However, for ball screw drives with heavier tables or a longer stroke, the variation ma become significant and a gain scheduling method ma be required, as implemented in [7][34].

25 3 Figure 3.5: Variation of sstem FRF with table position. 3.. Pole-Placement Feedback Design The equations of motion in Equation (3.) can also be represented in state-space: Cz B B B Az z 3, x x d d u (3.3) Where 3,,, ) ( ) ( m m m m b c m c m k m k m c m b c m k m k B B B A and also in transfer function matrix form: 3 3 d d u G G G G G G x x (3.4)

26 4 where s a s a s a s m m k s b c m s d x G s a s a s a s m m k cs d x G u x G s a s a s a s m m k cs d x G s a s a s a s m m k s b c s m d x G u x G ) ( ) ( and k b b a k m m c b b b b a c m m b m m b a ) ( ) ( ) ( ) ( 3 The states to be controlled are the position and velocit of the rotating and translating components. The position values are practicall measureable, and the velocit estimates can be obtained using backwards Euler approximation, s k k k s k k k T x x x T x x x,,,,,, ˆ, ˆ (3.5) It is also desirable to eliminate stead-state positioning errors on the nut side due to unmodeled disturbances and dnamic effects such as the motion loss in the preloaded nut [7], which cannot be captured with the linear model in Figure 3.3. Thus, the integrated table position, dt t x x i ) (, is also considered as an additional state to be controlled, resulting in the state vector to become, T i x x x x x z (3.6) The feedback controller is designed principall through the use of pole-placement control (PPC) [3]. The objective is to increase the dnamic stiffness of the ball screw around the st axial mode, while also realizing a sufficientl high tracking and disturbance rejection bandwidth for rigid bod dnamics. The model has 5 states; hence, 5 pole locations are required. Two of the poles ) (, p are placed to speed up the deca of axial vibrations without altering their oscillation frequenc. This is similar to the idea of low authorit LQG vibration control []. A change in the vibration frequenc would be more costl in terms of control effort, since these poles originate from the drive s mechanical structure. Hence, onl the real component is shifted further to the left in the s-plane as,

27 p, jd, where (3.7) In our setup, could be increased up to 4. The remaining three poles, which relate to the rigid bod motion interacting with the servo controller, which has integral action, are placed at a reasonabl high frequenc which will ield good disturbance rejection and tracking without significantl jeopardizing the stabilit margins. The were chosen to have a pole frequenc of complex with. 7 damping. Hence, 5 Hz with two of the poles being p p 3,4 j 5 (3.8) The state feedback gain K K K K K K is computed to achieve x x v v i eig ( A B K) { p p5}. Ackermann s formula could be used [4], but standard algorithms in software such as Matlab can also be applied, giving more robust results. The open- and closed-loop pole locations are illustrated in Figure 3.6. Figure 3.6: Open- and closed-loop pole locations. 5

28 In practical implementation, additional dnamics due to torsional modes of the ball screw and the motor current loop cause noticeable discrepancies between the modeled and actual drive response after 5 Hz. These can result in serious stabilit problems unless their effect is attenuated, which is generall achieved using low-pass filters. However, since the torsional modes exhibit ver little variation with table position and remain constant over the prolonged use of the ball screw, the can be attenuated with notch filters in the form: s zkn, k s n, k G notch( s) (3.9) k s pkn, k n, k with n, 4 Hz ( z. 55, p.7 ) and n, 8 Hz ( z. 5, p.7 ). Notch filters cause smaller phase loss compared to low-pass filters, allowing a higher crossover frequenc to be achieved. High frequenc dnamics beond the first two torsional modes are attenuated with a low-pass filter with a pole frequenc of Figure 3. can be expressed as: lpf G fp = Hz. The combined transfer function of the filter pack shown in lpf ( s) Gnotch( s) s (3.) lpf 3..3 State Trajector Generation and Model Inversion Following the feedback design, feedforward terms are designed to improve the command tracking. Since the command x corresponds to the desired table motion x ), the flexibilit of the ball screw must be r considered when generating the motion commands for the ball screw s rotar motion. This is achieved b the rotar command generator, G r (s) in Figure 3.. Solving Equation (3.) for x results in: ( r cs k x ( s) x ( s) d( ) m s ( b c) s k s (3.) If d is ignored, G r ( s) x x becomes, G r ( s) m s ( b c) s k cs k 6 (3.)

29 which determines the necessar rotational motion of the screw to accommodate the desired table motion ( x r ). Additional feedforward action is provided b inverting the plant dnamics G from the control signal input (u) to the table linear position ( x ). Hence, the model inversion filter G ff in Figure 3. becomes: G ff 4 3 mm s as as a3s ( s) (3.3) cs k The denominator coefficients a, a and a 3 have been defined in Equation (3.4). In real-time implementation, G ff and dela terms to make them causal Trajector Pre-filtering G r were discretized using pole-zero matching [4] and adding the necessar When the ideal plant and s-domain controllers are considered, the control signal can be expressed as: u( s) G K G K K ( s) ( K K s) G ff r x v fp xr ( s), where KG KG K( s) ( K x Ki / s Kvs) G (3.4) fp B substituting the expression for u in Equation (3.4) into the expression for G from Equation (3.4): x G G G ff G KGrG KG u K G K G x r (3.5) It can be seen that the overall tracking transfer function, from the trajector command to the table translational motion, simplifies to x x r. However, due to computational delas, sampling, and variation of the actual drive FRF from the ideal model, the tracking transfer function will be distorted. As a result, artifacts correlated to the velocit, acceleration, jerk and snap commands ma appear in the tracking error profile, as was experimentall observed and presented in Figure 3.7. To address this problem, a new tpe of trajector pre-filter has been designed, as shown in Figure 3.. 7

30 Figure 3.7: Tracking error profile with velocit, acceleration, jerk and snap artifacts. The actual distorted version of G e G track x / x G r will be referred to as track. The error transfer function,, is defined such that the observed translational tracking error can be obtained as e G e x r. As can be seen from Figure 3.7, the tracking error appears to be correlated with the velocit, acceleration, jerk and snap profiles through a quasi-static (stead-state) relationship. Hence, expanding the Maclaurin series for G e (s) around zero frequenc ( s ) where ields: 3 ( IV ) 4 Ge ( s) Ge () Ge () s Ge () s Ge () s Ge () s (3.6)! 3! 4! G e, Ge, G e and IV G are the first four derivatives of G e with respect to s. Since e G track has been designed to ield zero stead state error to position commands, G e ( ). Also, since there are no integrators in G e, the derivative terms will all be finite at s. Thus, G e can be estimated for low frequencies b identifing the first four nonzero terms of Equation (3.6), Gˆ e 3 Kvels Kaccs K jerks Ksnaps (3.7) 4 The coefficients Kvel G e (), K acc G e ()!, K jerk G e () 3!, and Ksnap G e () 4! are computed using the Least Squares technique to allow the prediction of e as a function of the filtered commanded kinematic profiles: 8 (IV )

31 e ˆ K ( IV ) vel x rf Kacc xrf K jerk x rf Ksnapxrf (3.8) Filtering is used, as shown in Figure 3., in order to avoid numerical problems with high order derivatives. The low-pass filter, G lpf, which is emploed multiple times, was designed to be st order with a pole frequenc of 8 Hz. The filtered velocit, acceleration, jerk and snap commands are obtained as, x x x x rf rf rf ( s) G ( s) G ( s) G ( IV ) rf lpf 3 lpf 4 lpf ( s) G ( s) x ( s) ( s) x( s) ( s) x ( s) 5 lpf ( s) x ( IV ) ( s) (3.9) A single tracking experiment, with the gains K vel, K acc, K jerk, and K snap set to zero, is sufficient to identif the filter coefficients. The resulting model allows adequate prediction of the tracking error, shown in Figure 3.7. If the commanded trajector is offset b the predicted error, the table position becomes, Using this result, the tracking error is now, x ' ˆ ˆ ) r xr Ge xr ( Ge xr (3.) x ˆ ) Gtrackx' r Gtrack( Ge xr (3.) e x x r r ( G ( G x G e track track G track ( Gˆ ) x G track e e Gˆ ) x r r Gˆ ) x e r (3.) For low frequencies, assuming that the approximation G ˆ G holds: e e 9

32 e Gtrack) e r e ( G x G x (3.3) Equation (3.3) shows that the tracking error is reduced for frequencies where G. In particular, this effect is realized at low frequencies where G track is designed to be close to one ( G e ). For example, if G e ( j )., the theoretical command following error at frequenc will be reduced further b a factor of due to the addition of the trajector pre-filter. r e 3.3 Frequenc Domain Analses Although the sstem has two outputs, the fact that there is onl one controlled input allows SISO-tpe frequenc domain analses to be carried out. The following frequenc domain concepts are used throughout the controller design, tuning and evaluation steps. Loop Transfer Function Given the plant, G (s), and the feedback controller, K (s), the loop transfer function is L( s) G( s) K( s). The bandwidth of a closed-loop sstem can be approximated b the crossover frequenc, which is frequenc at which L ( j). Nquist Stabilit Criterion The Nquist criterion provides a method for determining the closed-loop stabilit of a sstem. For a stable open-loop sstem, if the locus of the loop transfer function, L (s), plotted from s j to s j makes no net encirclements of the point s, the closed-loop sstem is stable. Sensitivit Function The sensitivit function is S ( j) ( L( j)). The magnitude of the sensitivit function corresponds to the inverse of the distance from the Nquist plot to the critical - point. Thus, the sensitivit function is a useful measure of stabilit margins, and in practice, it is desirable to keep its maximum value, M S max S( j) for frequencies at which S ( j). S, below Also, the controller activel rejects input-level disturbances In the analses, three cases are considered:

33 (a) ideal modeled plant and continuous time (s-domain) controller, (b) experimentall measured plant and s-domain controller, (c) experimentall measured plant and discrete time (z-domain) controller. The discretized version of the control law includes the phase dela effects of the zero-order hold and a full sample (i.e., worst case) computational dela, at a sampling frequenc of khz. Considering Figure 3.8, the loop transfer function for the mode compensating pole-placement controller (MC-PPC) can be expressed as: L KG KG (3.4) K and K were defined in Equation (3.4) and G and G were defined in Equation (3.4). Figure 3.8: Block diagram manipulation for assessing closed-loop stabilit. In order to demonstrate the effectiveness of the developed MC-PPC, it has been compared to the wellestablished P-PI position-velocit cascade controller that is commonl used on feed drives in industr. The P-PI scheme is shown in Figure 3.9.

34 Figure 3.9: P-PI position-velocit cascade control scheme with velocit and acceleration feedforward terms. To allow for a fair comparison, the MC-PPC and P-PI controllers were tuned to possess similar stabilit margins using frequenc domain analses. Both controllers have phase margins of PM 3 and maximum sensitivit values S max These stabilit margins are slightl more aggressive than commonl used values, favoring tracking accurac over robustness in order to evaluate the best possible performance that could be achieved with both controllers. In practical implementation, both controllers can be detuned as required in order to ield better stabilit margins. The filter pack, feedforward friction compensation are implemented in an identical fashion for both controllers. The loop transfer function for the P-PI controlled sstem can be found to be: G fp, and the K pvs Kiv L G fp ( Gs GK px) (3.5) s B inspecting the loop magnitude plots in Figure 3.ab, the frequenc domain results suggest that the MC-PPC will have superior tracking and disturbance rejection, as it has significantl higher low frequenc gain and achieves higher crossover frequencies of and 7 Hz, compared to 7 and 34 Hz

35 for the P-PI. The nd crossover frequenc with L ( j) indicates the frequenc range up to which the controller is activel damping the axial vibrations. Figure 3.: Loop magnitude of (a) MC-PPC and (b) P-PI; Nquist plot for (c) MC-PPC and (d) P- 3 PI. The sensitivit plots for the two controllers are shown in Figure 3.. Interestingl, the MC-PPC peak sensitivit value increases from.35 with the s-domain controller to 3.4 for the discretized version. Thus, a higher sampling frequenc beond khz could ield a significant improvement in the stabilit margins for this controller. On the other hand, the P-PI peak sensitivit onl increases from S max = 3.4 for the continuous time controller to 3.5 for the z-domain controller. This suggests the performance is

36 primaril limited not as a result of the effect of sampling, but rather due to the control structure which considers onl the rigid bod plant. Also, S for -34 Hz for the MC-PPC, compared to onl -46 Hz and -46 Hz for the P-PI controller. This indicates a larger disturbance rejection bandwidth for the MC-PPC. Figure 3.: Sensitivit magnitude for (a) MC-PPC and (b) P-PI. The open-loop transfer function from table disturbances, d (e.g., cutting forces), to the table position, x, is given in Equation (3.4) as G 3. The closed-loop disturbance transfer function for the sstem controlled b the MC-PPC can be obtained as: G dx G 3 K ( G G G G) 3 L 3 (3.6) For the sstem controlled b the P-PI controller, the disturbance transfer function is: G dx G 3 G fp ( K pvs Kiv)( GG3 G3G) L (3.7) B analzing the disturbance transfer functions in Figure 3., it can be seen that active damping ields improved disturbance rejection. The maximum negative real value of the disturbance transfer function is.83 rad/v at 45 Hz for the MC-PPC and.4 rad/v at 6 Hz for the P-PI. This indicates an improvement in the drive s immunit to machining chatter vibrations [3]. 4

37 Figure 3.: Disturbance transfer functions for MC-PPC (left) and P-PI (right) considering ideal plant. The ideal closed-loop tracking transfer function has a unit gain and zero phase. As can be seen in Figure 3.3, the feedforward terms significantl improve the tracking performance of the drive. The inertia and viscous friction compensation flattens both the gain and phase up to around Hz. However, discretization and computational dela cause distortion in the phase and, to a lesser degree, the magnitude. The pre-filter helps to mitigate this effect, reducing the phase lag up to ~ Hz and bringing the magnitude close to unit up to ~8 Hz. 5

38 Figure 3.3: Command tracking with MC-PPC (left) and P-PI (right) with experimentall measured plant 3.4 Experimental Results The performance of the MC-PPC was compared to the P-PI controller in experiments. The superior disturbance rejection capabilit of the MC-PPC can be seen in the results of rope snap (step disturbance) tests. As shown in Figure 3.4, the MC-PPC has a lower peak displacement due to its higher control bandwidth, and a much better rejection of the low-frequenc disturbance as seen in the faster return of the rigid-bod to zero displacement. The deca of the 43 Hz vibrations is also much quicker, as the MC-PPC has a 5 percent settling time of.3 seconds, compared to.63 seconds for the P-PI controller. B comparing the amplitude spectral densities [] in Figure 3.4b, it is seen that the MC-PPC step disturbance response has smaller low-frequenc amplitudes, as is expected given the larger low frequenc gain seen in Figure 3.a. The peak at 43 Hz corresponds to the pitching mode of the table, as discussed in Section

39 Figure 3.4: (a) Table displacement and (b) amplitude spectral densit of table displacement for the MC-PPC and P-PI in Rope Snap Test High speed tracking tests at, 5 and mm/s were conducted in order to validate the performance of the controller. Each test was repeated 5 times, and average results are shown in Figure 3.5abc. The MC-PPC significantl outperforms the P-PI both during constant velocit and during acceleration transients as indicated b the lower RMS and maximum errors in Table 3.. Furthermore, the controller performance was verified in metal cutting tests. Tests were performed at mm/s while performing a slotting operation with a 4 flute 5.4 mm end mill in 665 aluminum. The spindle speed was 35 rpm and depth of cut was. mm. Average results from the 5 tests are shown in Figure 3.5d. The tool enters the cut at around.6 seconds. At ~. seconds, as the end mill exits the cut and the last chip breaks free, the sstem is subjected to a step disturbance. The response is similar to Figure 3.4a. 7

40 Figure 3.5: Tracking error, control signal, and FFT of tracking error for (a) mm/s, (b) 5 mm/s, (c) mm/s and (d) 5.4 mm slotting at mm/s in 665 aluminum. 8

41 Table 3.: Maximum and RMS errors during tracking and machining experiments MC-PPC P-PI Velocit (mm/s) Max. (um) RMS (um) Max. (um) RMS (um) * *Machining experiment 3.5 Conclusions This chapter has studied the use of the pole-placement technique to achieve active vibration damping, as well as high bandwidth disturbance rejection and position control in ball screw drives. Motivated b a simple and phsicall intuitive model, two of the poles are placed to speed up the damping out of axial vibrations, while the other three represent the equivalent rigid bod dnamics coupled with the control law which possesses integral action. A new tpe of trajector pre-filter is also presented, which is easil tuned b conducting a single tracking experiment. The pre-filter mitigates the imperfections in the drive s tracking function due to sampling, computational dela effects and the discrepancies between the actual drive response and its assumed model. The MC-PPC design and pre-filter are combined together inside the proposed scheme, which has been validated to provide superior command tracking and disturbance rejection compared to the mainstream approach of appling P-PI position-velocit cascade control. Comparison between the two schemes has been performed in frequenc domain analses, as well as step disturbance (rope snap) tests, machining and high-speed tracking experiments. The MC-PPC, on average, provides a 4-55% reduction in the maximum, and a 55-65% reduction in the RMS values of the positioning error compared to P-PI. The MC-PPC also provides a significant bandwidth improvement and dnamic stiffness increase around the st axial mode. 9

42 Chapter 4 Control of a Linear Motor Driven Gantr 4. Introduction Direct drives provide substantial improvement in the dnamic response and accurac over conventional ball screw drives [8], while the gantr configuration gives the potential for an extended work area while preserving a certain level of positioning accurac. However, due to the absence of gearing between the motor and load, the drives are subjected directl to disturbance forces and an variations in the inertia, due to changes in the machine configuration or workpiece mass. Also, the gantr configuration tpicall results in at least one of the axes being actuated b two parallel drives where crosstalk is inevitable. This presents a significant challenge in snchronizing the axes while achieving high control bandwidth and ensuring stabilit. One mainstream solution is to tune the two actuators to ield an identical dnamic response, preventing one motor from falling behind the other during trajector tracking. Although this is simpler and more reliable than master-slave tpe interlocking [3], it has limitations, particularl when there is strong dnamic coupling through vibration modes between the two actuators. In feed drive control, the objective is to realize a high bandwidth in order to obtain the most favorable command following and disturbance rejection properties [9]. The control bandwidth can be significantl limited b the structural dnamics of the drive and machine tool. Recentl, a new generation of controllers that activel compensate for structural dnamics using linear motors or ball screw drives has been developed, as reported in [][9][3][34]. These controllers enhance both dnamic stiffness as well as positioning accurac. Appling similar techniques to gantr tpe motion stages presents a greater challenge due to the increased complexit of having to consider the dnamic interactions between the different actuators. Direct drive motors are well suited for appling active vibration damping due to their short response time. However, the are also more susceptible to the dnamic coupling problem encountered in gantr stages. In literature, Teo et al. [35] have developed a dnamic model for an H-tpe gantr, considering onl rigid bod dnamics and no vibration modes or stiffness values. The have reported successful experimental results on a ball screw driven gantr using adaptive control. 3

43 Weng et al. [4] have presented the concept of sensor and actuator averaging and shown its effectiveness in controlling flexible structures, such as beams and plates. One of the main advantages of this approach is the attenuation of structural resonances without detrimentall affecting the loop phase, which would normall be associated with the use of notch filters. In this paper, the sensor/actuator averaging has been adopted in a novel controller design for a T-tpe gantr. The gantr was developed and built at the Universit of Waterloo as a new x- stage concept for precision and micro/meso machine tools. The stage uses a single guidewa, which is particularl appealing as a simple design. A secondar linear motor at the unconstrained end of the gantr beam is used to provide both stiffness and damping enhancement. The control law has been designed to take into account the dnamic coupling between the actuators and to activel attenuate the first vibration mode encountered at 94 Hz on the gantr. The stabilit and robustness of the design is achieved through the use of multivariable frequenc domain analses. 4. Experimental Setup The experimental setup, shown in Figure 4. and Figure 4., has a T-tpe gantr and worktable configuration. Motion in the -axis is provided b the gantr, while the worktable moves along the gantr in the x-axis. The worktable is supported on a precision granite surface using a x (35 mm x 35 mm) vacuum preloaded (VPL) air bearing. The VPL uses positive and negative pressure to preload the table against the granite surface. The VPL provides both pitch moment and vertical stiffness, transmitting worktable and workpiece loads directl to the granite, rather than the gantr beam. The worktable is supported and aligned to the gantr beam using four rectangular air bearings. The -axis is driven b two linear motors and feedback is available on both the left (primar) and right (secondar) actuators via sinusoidal encoders with 4 um signal period. The design utilizes onl a single granite guidewa along the -axis, which simplifies the alignment and assembl of the stage. The gantr is constrained to the guidewa b four preloaded air bearings. The secondar linear motor provides stiffness enhancement and active attenuation of aw vibrations. The x-axis motion is generated b a third linear motor attached to the gantr beam, with feedback provided b an encoder similar to those on the -axis. 3

44 Figure 4.: Linear motor driven T-tpe gantr. Figure 4.: Schematic of the experimental setup. 4.3 Modeling and Identification 4.3. X-Axis Modeling and Identification During identification tests, the x-axis was observed to be reasonabl independent of the gantr excitation. When excitation was provided b the x-axis linear motor, the open loop acceleration frequenc response function (FRF), shown in Figure 4.3, was noted to be quite flat up to 5 Hz. As a result, the x-axis position response has been modeled as a double-integrator ( x ( m s )). Considering the low frequenc portion of the FRF, the control signal equivalent mass was estimated to be m x =.6977 V/(m/s ). Due to the use of air bearings, viscous and Coulomb friction were not observed. x Figure 4.3: Open-loop acceleration FRF for x-axis. 3

45 4.3. Y-Axis Modeling The -axis was observed to show strong coupling between the left and right hand actuation points, in particular due to aw vibrations that originate from the rigid bod rotation of the gantr assembl around the z-axis, constrained b the torsional spring-like effect of four air bearings preloaded against each other about the -axis granite guidewa. This has been illustrated in Figure 4.4 with a simplified dnamic model. Figure 4.4: Y-axis dnamic model. The simplified dnamic model in Figure 4.4 adequatel captures the inertial characteristics and aw motion of the gantr. B appling D Alembert s Principle, the equations of motion can be written considering the -axis position ( ) for the Center of Mass (CoM) of the beam, and the rotation angle ( ): m f J c k a a f M q C q K q ( t) ( ) t F q (4.) Above, m is the mass of the gantr assembl (including the table) and J 33 is its moment of inertia about its center of mass. c and k are the angular damping and stiffness coefficients, respectivel, provided b the air bearing assembl. a and a are the distance from the CoM to the left and right end points. f and f are the actuation forces provided b the left and right hand linear motors. Defining the kinematic transformation from right hand sides of the gantr: T [ ] to the displacements [ T ], measured b the encoders on the left and

46 34, a a T T (4.) Equation (4.) can be re-written in terms of T ] [ as:,, where : ) ( ) ( L k k L c c J m a J m a a J m a a J m a L t f t f k k k k c c c c m m m m q q q q q q M T K F K T C F C T M F M (4.3) L =484 mm is the span between the left and right motors. Using Equation (4.), ) (s and ) (s can also be expressed in the Laplace domain in terms of ) ( s f and ) ( s f. Following this, and appling the coordinate transformation in Equation (4.), the following transfer matrix relationships between the input forces T f f ] [ and measured endpoint displacements T ] [ can be obtained: k cs Js a ms G k cs Js a a ms G k cs Js a a ms G k cs Js a ms G f f G G G G, (4.4) Y-Axis Identification The parameters for the -axis model in Equation (4.3) have been identified b appling Least Squares fitting on the frequenc response data, following a methodolog similar to the one in [7]. Here, additional equalit constraints have also been incorporated for the total mass and viscous friction, which were estimated ahead of time. The equations of motion in Equation (4.3) can be written in the frequenc domain as: ) ( ) ( u u k k k k j c c c c j m m m m (4.5) which can be rearranged as,

47 k ( ) ( ) m j c k m j c u k ( ) u ( ) m j c k m j c Gˆ u (4.6) The simplified model assumes that the MIMO plant G contains onl one mode. Assuming that the estimated model Ĝ closel approximates the real plant ( G ˆ G) in a particular frequenc range of interest which includes onl this mode, ˆ g( ) jh( ) g( ) jh( ) G ( ) G I, where G( ) g( ) jh( ) g( ) jh( ) (4.7) Above, g and h represent the measured real and imaginar components of G, G, G, and G. B separating the real and imaginar response terms from the dnamic sstem parameters, Equation (4.7) can be rewritten in parameter-regressor form ( Y ) b defining the parameter vector, : m m m c c c k k k T (4.8) and regressor matrix, (), at a particular frequenc, g h ( ) g h ( ) ( ) ( ) ( ) g h g h g h g h ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) g h g h ( ) ( ) ( ) ( ) h g h g ( ) ( ) ( ) ( ) h g h g h g h g ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) h g h g ( ) ( ) ( ) ( ) g g h ( ) h ( ) ( ) ( ) g h g g h g h ( ) ( ) ( ) h ( ) ( ) ( ) ( ) ( ) g ( ) h( ) g( ) h ( ) (4.9) Given the FRF measurements at a set of frequencies,,,, N, around the mode of interest ( 6 Hz, N 5 Hz), we obtain, 35

48 ) ( ) ( ) ( N N N N N N e e e Y (4.) The objective is to find that minimizes, ) ( ) ( Y W Y T J (4.) Above, W is a weighting matrix which allows the importance of the measurement at each frequenc to be specified. The weighting matrix has the following form: , I w w w w W j j N N N w (4.) In our estimation, all measurements for the frequenc range of interest (6-5 Hz) were weighted equall ( j w ). The total control signal equivalent mass ( m =.7 V/(m/s )) and its distribution on the left ) / ( m m and right hand ) / ( m m sides of the gantr were estimated b correlating the control signal (i.e., force demand) during controlled tracking tests with the commanded acceleration profile using Least Squares. The variation of the mass distribution with x-axis position is shown in Figure 4.5.

49 Figure 4.5: Mass distribution variation with x-axis location. The known mass distribution ( m m m; m m m) identified during these controlled tracking experiments, a zero external viscous friction condition ( c c c ), and smmetric stiffness ( k k k) and damping ( c c c) have been incorporated into the estimation as constraints using Lagrange multipliers. The constraints are stacked together in matrix form as L, m m m m 3 3 m 3 c 3 c 3 3 c 3 k 3 k L k (4.3) The objective function including the constraints is: J' ( Y ) T W( Y ) T ( L ) (4.4) 37

50 B setting gradients with respect to the parameters and the Lagrange multipliers ( ) equal to zero T ( J '/ and J '/ ), the least squares solution for [ ] can be obtained as, T T W L T WY A L A b b (4.5) This gives the parameters, which allows the control signal equivalent inertia ( J J mx ( d ) ), torsional damping (c) and stiffness (k) to be calculated using the relations in x Equation (4.3). The identified parameters are, J J m x m x ( d x).6977[v/(m/s, where J.434[Vms )]; c 8.7[Vms]; k 59. ], d.655[m], 3 [Vm] (4.6) In FRF measurements, shown in Figure 4.6, the lowest resonance due to aw vibrations was noted at 94 Hz. The vibration mode registered at 38 Hz is attributed to the bending of the gantr beam about the z- axis. The mode at 3 Hz could not be attributed to either pitching or bending of the gantr, and requires more investigation. As the x-axis location of the table varies, the inertia distribution on the left and right hand sides will also change. As a result, the vibration modes were observed to shift in frequenc b less than ±6%. Figure 4.6: Y-axis acceleration response. 38

51 The model in Equations (4.) and (4.6) predicts the dependenc of the rotational inertia, hence the aw mode, on the x-axis position of the table. At x = mm, the first mode is predicted to be at 94.3 Hz with 3.4% damping. The model for the table at different positions has been shown in Figure 4.7, and is in good agreement with the experimental FRFs. Figure 4.7: Model identified for -axis position FRF with table at x = -.5 m (left), x =. m (center) and x = +.5 m (right) Certain structural modes were also confirmed using impact hammer testing, with feedback from the linear encoders. The hammer test results are shown in Figure

52 Figure 4.8: Impact hammer test results. B varing the hammer input location along the length of the gantr, the shape of the aw and bending modes can be seen. At 94 Hz, the excitation points on the same side of the CoM are moving in phase, and are 8 degrees out of phase with points on the opposite side. The magnitude of the resonance at each point is proportional to the distance from the CoM. This indicates a rigid-bod rotation about the center of mass. At 38 Hz, points A and D are moving in phase, while B is out of phase. Point C appears to be close to a node of this mode. This is attributed to the first bending mode of the beam. 4

53 Also, b moving the excitation input from the bottom to the top of the gantr, an additional mode at 6 Hz can be observed. The top and bottom of the gantr are moving out of phase, indicating that the gantr is pitching about the x-axis. The excitation point B is close to the neutral point of this mode, which is assumed to be the gantr CoM. Similarl, the pitching mode is not observed in the FRF s when excitation is provided b the motors, since the motors were verticall positioned to appl their actuation force ver close to the CoM. Overall, the -axis modeling and identification results indicate that the most significant obstacle to achieving high control performance is the aw vibration mode at 94 Hz. This mode dramaticall limited the control bandwidth to around 6 Hz when stabilization was attempted using independent PID controllers for both actuators. Therefore, this mode has been analzed and taken into account in the controller design, in order to further improve the -axis performance, in the proceeding sections. 4.4 Controller Design 4.4. X-Axis Controller Design The x-axis controller was designed using classical loop shaping methodolog []. First, a desired crossover frequenc ( c ) and phase margin (PM ) are chosen. The highest crossover that could be achieved while maintaining adequate stabilit margins was c = 3 Hz. The phase margin was chosen to be 35. In order to eliminate stead-state errors, integral action was also added. The integral term takes the form ( T i s ) s. T i is designed to give a phase lag of stabilize the x-axis, the required phase lead at c is found as, PI = - at the crossover frequenc. To lead 8 PM Gx ( jc ) PI 6 (4.7) Above, Gx ( jc ) is the phase angle of the x-axis position FRF at c. Since the required lead cannot be provided b a single lead filter (>9 ) or a derivative term, two lead filters are used, each of which generates +63 phase shift at c. This requires each filter to have a lead ratio of ( sin 63) /( sin 63) 4. []. The filter pole is placed at a c and the zero at c / T d /. The gain K is adjusted to make the loop magnitude unit at c. Hence, the double lead and integral compensator becomes: 4

54 ( Ti s ) ( Td s ) C( s) K s ( s / a ) (4.8) The loop transfer function, L ( j) G( j) C( j), and the Nquist plot are used to inspect the crossover frequenc and phase margin. The magnitude of the sensitivit function, S ( j) /( L( j)) is the inverse of the distance from the Nquist plot to the critical - point, and so can also be used as a measure of stabilit [3]. A st order low pass at Hz, and Notch filters in the form N( s) ( s NN s N ) /( s DDs D ) at 86 Hz ( N =.6, D =.7), 33 Hz ( N =.3, D =.7), and 44 Hz ( N =., D =.7), were inserted to attenuate high frequenc modes and mitigate undesirable peaks in S ( j). When the design was complete, the specifications for c and PM were achieved. The controller is active where S., which indicates an improvement over the open-loop disturbance response. This range was up to 98 Hz. The continuous-time controller has been discretized using pole-zero matching with a sampling frequenc of loop magnitude, Nquist and sensitivit plots are shown in Figure 4.9. T s 5 khz. The corresponding Figure 4.9: X-axis loop gain (left), sensitivit (center) and Nquist plot (right). Feedforward inertia compensation and a trajector pre-filter were added to further improve command tracking. The control effort for inertia compensation is, u ff mxxr (4.9) 4

55 An additional trajector pre-filter was designed to remove an remaining correlations of the velocit, acceleration, jerk, and snap (4th derivative) commands from the tracking error, b offsetting the position commands accordingl: x ' x r r K vel x rf K acc x rf K jerk x rf K ( IV ) snapxrf where x x x x rf rf rf ( s) G ( s) G ( s) G ( IV ) rf lpf 3 lpf 4 lpf ( s) G ( s) x ( s) ( s) x( s) ( s) x ( s) 5 lpf ( s) x ( IV ) ( s) (4.) are low-pass filtered versions of the commanded velocit, acceleration, jerk and snap. The low-pass filter was designed in the form G lpf ( s) /( s ), where c 8 Hz. The pre-filter gains were identified using Least Squares from the results of a single tracking experiment. c c 4.4. Y-Axis Controller Design Due to the strong dnamic coupling between the left and right hand actuators, particularl through aw vibrations at 94 Hz, design of the -axis controller proved to be more challenging. The overall scheme is shown in Figure 4.. The principal regulation function is achieved b controlling the CoM motion for the gantr, b using sensor and actuator averaging and loop shaping. Active vibration damping is added to improve the dnamic stiffness against external disturbances, such as machining forces. Actuator level integral action is also added to eliminate stead-state positioning errors at the servo level. To enhance the command tracking, feedforward inertia compensation and a trajector pre-filter are included into the scheme. Furthermore, a filter pack comprising of notch and low-pass filters ensures that unwanted sensitivit peaks, which ma lead to poor robustness or instabilit, are avoided in the feedback loop. 43

56 Figure 4.: Y-Axis (gantr) control scheme. In the following sections, the function and design of each of the components will be explained. One of the most crucial issues with the MIMO controller is guaranteeing stabilit with adequate margins. For this purpose, frequenc domain analses were conducted during the design iterations, using several concepts from multivariable control: Singular Value Decomposition (SVD) Singular value decomposition (SVD) is a useful matrix factorization which allows several SISO control concepts to be intuitivel extended to MIMO sstems. The SVD theorem states that an matrix, can be decomposed into a product of three matrices: nm nn nm H mm A U V (4.) A nm, where V and U are orthogonal rotation matrices which specif the input and output directions, respectivel. diag{,, n } is a unique scaling matrix containing the singular values, i, of A, in descending order. The maximum singular value of A is denoted as { A } and gives the largest matrix gain for an input direction. Likewise, the minimum singular value is corresponds to the smallest matrix gain for an input direction. {A} n and Loop Transfer Matrix Considering the experimentall measured open loop position frequenc response matrix G ( j), and the feedback controller response K ( j) (which here can be computed analticall from Equation (4.3)), the multivariable loop transfer function matrix is defined as L ( j) = G ( j) K( j). L ( j) shows the 44

57 combined amplification of the controller and plant. In particular, the feedback is active where { L ( j)}. MIMO Nquist Criterion According to the MIMO Nquist Stabilit Criterion, stabilit is guaranteed when the locus of det{ I L( j)} makes no counter-clockwise encirclement of the origin in the complex plane and does not pass through the origin [3]. MIMO Sensitivit Function The MIMO sensitivit function is defined as S ( j) ( I L( j)). M S max ( S ( j)) S is an important indicator of stabilit margin, which in practice should not be much larger than M.5.5. It is also noted that for frequencies where ( S ( j)), the controller is activel S rejecting disturbances in all input directions. Disturbance Transfer Matrix The disturbance transfer matrix is analzed to determine the disturbance rejection capabilit of the sstem. For the open loop case with our gantr model, it is equal to the MIMO plant, G ( j). When feedback is added, it becomes ( I L( j)) G( j) S( j) G( j). In particular, the upper singular value of the disturbance transfer matrix is examined, as it indicates the maximum possible amplification of a disturbance acting on the gantr drive sstem. Considering that a and a locate the CoM from the left and right actuation points, the mass distribution for the two sides of the gantr can be expressed as: m / m a / L, m / m a / L, where : a a L (4.) When the left and right encoder readings are averaged proportionall to their mass ratios (sensor averaging), the CoM displacement can be computed as: 45

58 T a / L / L a / L / L (4.3) On the other hand, defining the total actuation force as f T f f, if the total actuation force is distributed to the left and right hand drives proportionall to the mass the actuate (i.e., f f T ), the following averaged transfer function ( G (s) CoM displacement ( ) and total actuation force ( f T ): f f T and ) determines the relationship between the ( G G) ( G G) f T G ( s) (4.4) B substituting G, G, G, and G from Equation (4.4), the sensor and actuator averaged transfer function interestingl becomes G( s) /( ms ), in which the effect of the aw mode vanishes. From a feedback control point of view, this can provide a major advantage in increasing the bandwidth while avoiding the phase distortion associated with notch filtering, similar to the idea reported in [4]. The loop shaping filter Ds is designed to control the averaged transfer function G s derived in Equation (4.4). The loop shaping methodolog allows higher flexibilit in specifing the loop magnitude and phase, compared to more mainstream design techniques, such as P-PI position-velocit cascade or PID control. The feedback controller that realizes CoM position control through sensor/actuator averaging and loop shaping can be written as: u u, CoM, CoM D( s) Actuator Averaging Sensor Averaging e e ( s) ( s), e e r r (4.5) The loop shaping filter, D (s) is a lead filter designed to ield a 4 phase margin at the crossover corresponding to the rigid-bod motion of the CoM, Td s D( s) K s / a (4.6) 46

59 In this application, this crossover frequenc was chosen as c 6 Hz. The lead ratio, a / c c /(/ Td ), must be designed iterativel based on the phase associated with the averaged transfer function, G j ) and filter pack h( j c ) at the crossover frequenc. The design of the filter ( c pack is detailed later in this section. In practical implementation, the mass distribution estimates will not be exact. In addition, cutting or other external process forces ma still excite the aw mode vibrations. Hence, as a further measure, active damping is injected into the loop b emulating the effect of a torsional damper: u u, AD, AD s ( s) ( s) s( s( ) ) (4.7) Noting that ( ) / L, a / L, and a / L, b checking the characteristic equation after substituting Equation (4.7) into Equation (4.) as f u, AD and f u, AD, it can be seen that active vibration damping effectivel increases the torsional damping coefficient b aa, with each actuator contributing equall to this damping increase: ( Js cs k) ( s) aas aas [ Js ( c aa) s k] ( s) Actuator Contributi on Actuator Contributi on (4.8) Here, was designed to provide 59% increase over the existing damping provided b the air bearing assembl. Another advantage of the active damping is that it reduces the sensitivit, ( S ( j)), in the vicinit of the aw vibration mode at 94 Hz, making it possible to achieve more aggressive controller designs with higher bandwidths. In the current design, the active damping was necessar to avoid instabilit, as can be seen in both the frequenc domain analsis and time domain results in Figure 4.. When active damping is disabled, the sstem immediatel enters aw vibrations which are two orders of magnitude larger than the errors during high speed tracking tests with active damping. 47

60 Figure 4.: Contribution of active damping demonstrated in frequenc and time domain results. Tracking results are for a mm/s velocit, m/s acceleration, 5 m/s 3 jerk trajector. B averaging the sensor data, the situation ma arise where opposing servo errors at and give the false impression that the CoM error is zero (i.e., e e ). Thus, integral action has been implemented at the actuator level as follows: u u, i, i ki s e e ( s) ( s) (4.9) Aggressive values for k i were observed to excite low frequenc machine base vibrations, which, when interacting with the control law, resulted in vibrations around 36 Hz. This problem was predicted and avoided using multivariable stabilit analses conducted throughout the design, as seen in Figure 4.. Figure 4.: Maximum singular values of sensitivit function for various integral action gains (k i ) 48

61 Frequencies at which (S) >.5-.5 have been attenuated b placing notch and low-pass filters. As a result, h (s) was designed to contain notch filters at Hz ( N =.5, D =.7), 3 Hz ( N =., D =.9), 4 Hz ( N =.5, D =.), and 94 Hz ( N =.6, D =.75) and a nd order low-pass filter at 45 Hz. Since the active vibration damping results in a second crossover frequenc around c 4 Hz, a second lead filter ( lead 54 Hz, lead 7) was also inserted in the filter pack, to alleviate the corresponding peak in sensitivit. B combining the sensor and actuator averaged loop shaping, active damping, and integral action terms from Equations (4.6), (4.7), and (4.9), together with the filter pack h (s), the overall feedback controller can be obtained as: K( s) h( s) D( s) ki s s (4.3) To further improve the command tracking, feedforward inertial compensation and a trajector pre-filter were added as shown in Figure 4., in a similar manner to the x-axis design. The inertia compensation is in the form, u u ff ff mˆ mˆ mˆ mˆ r r (4.3) The trajector pre-filter offsets the commanded trajector independentl for each end of the gantr to remove correlations of the commanded velocit, acceleration, jerk, and snap (4 th derivative) profiles from the individual actuator servo errors: r r r r K K vel, vel, rf rf K K acc, acc, rf rf K K jerk, jerk, rf rf K K snap, snap, ( IV ) rf ( IV ) rf (4.3) The mass distribution terms and, the trajector pre-filters and the feedforward inertia compensation have been gain-scheduled in terms of the x-axis position to account for inertia variations due to table motion, 49

62 M M( xr ), ( xr ), ( xr ) (4.33) All filters were discretized using pole-zero matching with a sampling frequenc of 5 khz. When the design was complete, the peak sensitivit of.6 occurs at 4 Hz. The first crossover frequenc was at 5 Hz. The controller was also active in the range of 83-4 Hz, corresponding to damping of the aw mode. The MIMO loop magnitude and Nquist plots are shown in Figure 4.3 and the sensitivit function is shown in Figure 4.4. Figure 4.3: Y-axis loop transfer function upper and lower singular values (left) and MIMO Nquist plot (right). Figure 4.4: Upper and lower singular values of the sensitivit function for the proposed MIMO controller (left) and the SISO PID controllers (right). 5

63 When a decoupled PID controller with identical notch and low-pass filters was tuned to give a similar sensitivit peak (shown in Figure 4.4), the maximum achievable crossover frequenc was onl 6 Hz. Comparing the open- and closed-loop disturbances responses in Figure 4.5, when appling the MIMO control law the low frequenc disturbance rejection is improved up to 8 Hz, and in a wide frequenc band around the aw mode (65-5 Hz). The aw mode resonance is attenuated b a factor of ~5, with the vibration frequenc shifting from 94 Hz to 5 Hz. On the other hand, the SISO PID controllers onl provide improved disturbance rejection up to Hz, and times attenuation of the aw mode resonance, while shifting it to Hz, as seen in Figure 4.5. Figure 4.5: Singular values of open (G) and closed-loop (SG) transfer functions for the proposed MIMO controller (left) and independent SISO PID controllers (right). 4.5 Experimental Results The developed control law has been compared to the standard approach of using independent PID controllers in step disturbance (i.e., rope-snap), tracking and contouring tests. The improved disturbance rejection capabilit of the MIMO control law over the SISO controller can be seen in rope-snap tests, as shown in Figure 4.6. The tests consisted of appling a horizontal load of 57 N at the center of the worktable using a string and pulle sstem, and then cutting the string and monitoring the table motion. The table displacement was measured directl using a Renishaw RLE laser interferometer. 5

64 Figure 4.6: Experimental setup for step disturbance (i.e., rope-snap) testing With the MIMO controller, the table motion had a significantl lower peak displacement of 6.6 um, compared to 3.3 um with the SISO controller. The initial displacement of.7 um measured b the laser interferometer is a result of the mechanical flexibilit of the machine (approximatel 3.5 N/um). The amplitude spectral densit [] shows significant disturbance rejection improvement at low frequencies. The 3 Hz vibrations seen in the MIMO response correspond to the machine base vibrations. The controller can be detuned to reduce these vibrations, or new machine supports can be designed to mitigate this problem further. Overall, the disturbance rejection improvement provided b the proposed MIMO controller over the mainstream method of appling independent PID s is ver clear. 5