An Analysis on N-Forcer

Transcription

1 An Analysis on N-Forcer Irmak Aladagli DCT Traineeship report Coach(es): Supervisor: Dr. ir. R. L. Tousain Dr. ir. G. Angelis Prof. dr. ir. M. Steinbuch Technische Universiteit Eindhoven Department Mechanical Engineering Dynamics and Control Technology Group Eindhoven, June, 2009

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3 Abstract The N-Forcer is a magnetically levitated positioning machine prototype developed by Philips Applied Technologies (Apptech). The motivation of producing such a magnetically levitated machine is the need of a stage that can precisely follow a set point by avoiding nonlinearities caused by mechanical contact forces and that can operate in vacuum. This can be and has been realized with many different methods. The interesting thing about the N-Forcer among all other realizations is that it achieves this by using relatively inexpensive off the shelf linear actuators. The N-Forcer is actuated using slightly modified off the shelf ironless linear servo motors (ILSM s). Typically, ILSM s consist of a magnet track and a forcer, composed of current carrying coils. The interaction between the magnetic field and the coil results in a unidirectional force. However, six degrees of freedom (DOF) actuation is required for the N-Forcer so that it can be levitated, propelled and aligned. The innovation by Apptech thus introduces the modifications that enables these single DOF actuators to produce three independent orthogonal forces. Modification of the ILSM s consists of two steps. Firstly, the forcers are repositioned with respect to the magnetic filed. Secondly, the number of forcers per magnet track is increased, hence the name, N-Forcer. When the stage is actuated using two of these modified three DOF actuators, orthogonally oriented with respect to each other, one attains the required six DOF actuation. The N-Forcer, to be analyzed in detail, thus introduces a relatively cheap and supplier independent way of producing a magnetically levitated pick and place machine. This report presents the results of author s 13 week internship work on the control issues of the N-Forcer. During this internship, properties and actuation principles of the setup and K-factor compensation techniques were analyzed and experiments were performed in order to improve the performance of the system. As a result of these experiments, new compensation elements were introduced, which resulted in an improvement by a factor of 5. Tracking error at a constant speed of 120mm/s was decreased from 2µm to 400nm and from 5µm to 1µm for motion profiles having accelerations of up to 12 m/s 2. The error signal at standstill is about 40 nm at 50 Hz, which implies that a major source of the remaining error in the system is the amplifiers. In addition to improving the performance of the system, the new compensation revealed the error pattern related to the imperfect magnetic field of the ILMS s, allowing K-factor compensation techniques to be applied in the future. Further work is going on to achieve accuracy in the order of nanometers.

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5 Contents 1 Introduction Background Motivation and Goal of the Research Outline of the Report N-Forcer: The Setup Components of the Setup Actuation Principles Attaining 2-D Forces Forces on the Forcer Coils: Some Experimental Results Properties of Actuators Used in the N-Forcer Setup Measurement System The Controller K-factor Compensation K-Factor Compensation: Aim K-Factor in the ILSM K-Factor Compensation Algorithm Experimental Work On the N-Forcer Setup Directionality of Position Dependent Disturbance Force Compensation Table Tuning of the Feedforward Controller Robustness of Position Dependent Force Compensation Table Against Velocity Difference of the Error Pattern of the First Scans Analyzing the MIMO System Conclusion and Recommendations 36 A Further Explanations 37 A.1 Vibration Isolation A.2 Excitation of the Table by the Moving Chuck A.3 Delay A.4 M-File for Deciding on Best Acceleration Feedforward and Delay Correction B Data Sheets 43 B.1 ILSM Data Sheets

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7 Chapter 1 Introduction 1.1 Background One of the most important research branches actively investigated at Philips Applied Technologies (Apptech) is high performance positioning systems such as motion systems of wafer scanners. Wafer scanners are lithography machines used in the mass production of integrated circuits (IC s). These machines use a light source through a reticle that shapes the light, to illuminate a photo-sensitive layer on the silicon wafer. More detail on the history and working principles of wafer scanners can be found in [1] and [2]. Wafer scanners need to be as fast as possible to increase the throughput and as precise as possible to ensure the quality of the final product. One of the factors determining the precision, that is the size of the smallest possible printing, is the wavelength of the light from the light source. Light with short wavelength makes the illumination of finer details possible. Due to this, ASML, the market leader in the production of wafer scanners, plans to use Extreme Ultra-Violet light (EUV), which has very short wavelength, in its next generation wafer scanners. However, EUV is the most highly absorbed component of the electromagnetic spectrum, requiring high vacuum for transmission (see [3]). From designer s point of view, these demands mean that the reticle and the wafer should be positioned as fast and as precise as possible with respect to each other, in vacuum. From a control engineer s point of view, the machines should be designed as linear as possible to enable good performance by a control system. To achieve such a linear system, great measure should be taken to avoid disturbances and nonlinearities, whenever it is possible to do so. A big source of disturbances and nonlinearities in a motion system is contact forces like friction, mechanical backlashes, guide surface nonlinearities, etc. Thus if mechanical contact is avoided, a more linear system is achieved. For current wafer scanners, this is achieved by using air-bearings. However air-bearings are not suited for vacuum environment. Neither are mechanical bearings for that matter as the lubricant would contaminate the environment. Magnetic bearing and linear actuation is proposed as a solution to this problem and is even being employed for the new generation scanners (see [2]). A schematic of the components of a wafer scanner can be seen in the figure 1.1. As can be seen in this figure, light from the source is first shaped by a reticle that is being carried by the reticle stage. The light through the reticle is then shone on the silicon wafer after passing through the lens system. The wafer is carried by the wafer stage. The light source and lens system of the wafer scanner are fixed while the reticle stage and the wafer stage are actuated by two independent motion systems, both controlled in 6-DOF. The positioning of the wafer stage in itself involves two steps, the long stroke (LS) and the short stroke (SS). Traditionally, the 4

8 2 CHAPTER 1. INTRODUCTION Light source and light shaping Reticle stage with reticle containing the die pattern Lens with 4 reduction of reticle pattern to image on the wafer Wafer stage containing the wafer Figure 1.1: Schematic of the basic lay-out of a wafer scanner, obtained from [7]. Figure 1.1: A schematic representation of the components of a wafer scanner (from [2]) accurate 6-DOF SS positioning system (accuracy in the order of nm) sits on the less accurate 3-DOF LS (accuracy in the order of µm). The current actuator for the new generation LS is called a Synchronous Permanent Magnet Planar Motor (SPMPM). This type of motor uses permanent magnets arranged in a plane and coils to levitate and propel its load (see [2]. position Scan 1.2 Motivation and Goal of the Research Although the SPMPM is a possible actuator for the LS, research on cheaper and more flexible actuator led to the invention of a newstep actuator called the N-Forcer and the experimental setup that uses this type of actuators time called with the same name, hopefully without causing any confusion. Figure 1.2: Wafer with chips (left) and the step (bottom) and scan principle. In step lithography the position is constant during exposure, and in scan lithography the velocity is constant Figure 1.2: A photo of the N-forcer. N-Forcer, the setup as can be seen in figure 1.2, is a magnetically levitated positioning machine prototype developed at Apptech. It aims to accurately move a chuck on top of a frame without any contact forces. The fact that the chuck floats over the frame with no contact forces introduces many advantages like the elimina- 5

9 tion of nonlinearities caused by friction and mechanical backlashes, the ability to move without the necessity of a mechanical arm, etc. In addition, this system can be operated in vacuum environment, when needed. Although there are many magnetically levitated setups realized with different methods, N-Forcer has some interesting properties that make it a novel design. In contrast to other systems, N-Forcer is actuated with linear motors. These actuators are relatively cheap and supplier independent. However they produce forces in 1-D while the setup needs to be actuated in six degrees of freedom (DOF). Here comes the innovation. Apptech developed a way to easily modify these off the shelf actuators so that they produce 3-D forces. Thus, when 2 motors, oriented perpendicularly with respect to each other are used, system can be actuated in 6-DOF, within the tolerances of the system. The ultimate accuracy that is expected for this system is in the order of nm. The accuracy of the N-Forcer of course depends highly on how it is controlled. There are few challenges in the control of the N-Forcer. First of all, since the system uses ILSMs, it suffers from force ripple, i.e. position and load dependent disturbance forces. K-Factor compensation techniques are widely applied in iterative learning control (ILC) schemes to eliminate this kind of error and MSc research is already being carried out in this direction by J.M. Prevoo. However for the ILC algorithms to work, the error caused by the force ripples should be easily identified from the total error signal. This is only possible by eliminating other errors that constitute a big part of the total error. Thus the goal of my research can be divided into few main sub-goals. First sub-goal is to attain an understanding of K-factor compensation techniques through a theoretical study. Second sub-goal is to enable the application of K-factor compensation algorithms developed by J.M. Prevoo by identifying and eliminating errors that interfere with the identification of the K- factor related errors. This requires practical experimentation on the test setup and finding proper compensations for the related errors. 1.3 Outline of the Report This report consists of five chapters. First chapter gives background information on the context, motivation and the goal of the research. Second chapter gives a detailed analysis of the experimental setup. The information on the components, actuation principles, measurement system and initial control elements can be found in this chapter. Third chapter continues with the theory of K-factor compensation techniques. Fourth chapter contains experimental work on the setup. This chapter explains the experiments designed to find the different sources of errors and the measures taken to eliminate them. Since my work on the setup was a chronological improvement of the product instead of a structured study, fourth chapter is presented in a chronological order. I believe this is the best way to explain the reasons and results of my work. Finally fifth chapter includes the conclusions and recommendations for future work. Throughout the text, there are many references to appendices as I included most of my theoretical thinking here. Concluding the first chapter, I would like to reflect on my internship experience and thank my colleagues. I strongly believe that I gained valuable experience through this internship of 13 weeks as instead of working on a theoretical problem as I have been doing in the university for many years, I took part in the development of a product and faced the issues of this process. I would like to give my thanks to Professor Steinbuch for giving me the opportunity to such a good internship, to my supervisors Rob Tousain and to Georgo Angelis for their guidance through this internship and to fellow students Jan Maarten Prevoo and Duncan Denie for helping me with the experimental setup and for sharing their previous work and 6

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11 Chapter 2 N-Forcer: The Setup This chapter includes a detailed explanation of the N-Forcer. It starts with the physical properties of the setup and continues with the actuation principles, its measurement principles and its controller. 2.1 Components of the Setup The experimental setup is composed of three parts, the N-Forcer, the supporting table and the measurement system. The first part, the N-Forcer, consists of two main parts, a chuck and a frame. As depicted in figure 2.1, the chuck floats over the frame without any mechanical contact. Figure 2.1: Top: The chuck of the N-forcer floats without mechanical contact. Left: The chuck. Right: The frame. (figures from [4]) The chuck is driven in three dimensional space making use of Lorentz forces resulting from the interaction between magnetic field and moving charges. The actuator used in the N-Forcer, making use of these Lorentz forces, is called the Ironless Linear Servo Motor (ILSM), and is depicted in figure 2.2. The principle of actuation of this linear motor is to create a propulsion force between the permanent magnets attached to the frame and the current carrying coils attached to the chuck. In figure 2.1, the permanent magnets, arranged into magnet tracks, are depicted in red. The right picture shows three of such tracks while the actual setup also has the 8

12 fourth one that completes the symmetry. The current carrying coils, arranged into plates, called the forcers, are depicted in yellow. The actual setup has eight such forcers. Each group of two forcers move in one magnet track. Further explanation on the actuation principles will follow in the next section. The second part of the setup is the supporting table. N-Forcer setup sits on a suspended granite table to isolate it from ground vibrations. The stiffness of the suspension is very low as expected, so that it can function as an isolator of high frequency vibrations (see appendix A.1 for further explanation). This however causes some disturbance forces to act on the chuck for several reasons. First of all, since the table is not leveled with respect to gravitational acceleration, and since the measurement coordinate lies on the table, the horizontal force F x that is supposed to account for the acceleration of the chuck only, causes a direction dependent nonlinearity. That is, when chuck is moving in one direction, F x does work against gravitational acceleration. When the chuck is moving in the opposite direction, gravity helps this force by doing work on the chuck. Thus an offset is observed in measured F x while the chuck is moving in opposite directions with the same motion profile. Secondly, the motion of the chuck introduces vibrations of the table. An analysis of this disturbance can be found in appendix A.2. The granite table is fixed to a metal frame. This frame is not stiff enough, probably due to absence of diagonal elements between its legs. This introduces another disturbance with a very large amplitude. The natural frequency of the frame is about 12.5 Hz, and is easily felt by touching it. When the chuck operates with the air mounts off, the position error is dominated by a 12.5 Hz signal, in spite of the presence of the feedback controller. The amplitude of this error signal is about 1 µm, which is probably too large originally for the feedback controller to eliminate totally. When the air mounts are turned on on the other hand, this 12.5 Hz is not distinguishable in the power density spectral analysis. The third and the last part of the setup is the measurement system. The position of the chuck is measured by laser interferometry in all 6-DOF. The interferometry system rests on the same suspended table as the N-Forcer. Further analysis of the measurement system can be found in section Actuation Principles The N-Forcer is actuated using slightly modified off the shelf ironless linear servo motors (ILSMs). Typically, ILSMs consist of a magnet track and a forcer, composed of current carrying coils. The interaction between the magnetic field and the coil results in a unidirectional force. However, six degrees of freedom (DOF) actuation is required for the N-Forcer so that it can be levitated, propelled and aligned. The innovation by Apptech thus introduces the modifications that enables these single DOF actuators to produce three independent orthogonal forces. The following sections will explain the actuation principles of ILSMs and the modification by Apptech in detail Linear Synchronous Motors ILSM is a type of linear synchronous motor (LSM) that work using Lorentz forces, that is the force on a charge moving through a magnetic filed. A picture and schematic representation of the ILSM can be seen in figure 2.2. The ILSM has two parts, the magnet track and the forcer. Magnet tracks are composed of permanent magnets of alternating polarity, arranged into two walls. The forcer, which is a plate of coils, move in this track as seen in figure 2.2. The polarities of the magnet blocks alternate, to approximate a sinusoidal magnetic 9

13 Figure 2.2: Left: Off the shelf magnet tracks. Right: Magnets, arranged into two walls, have alternating polarities. Coils, represented by the red rectangles, move inside the magnet track. (figures from [4]) field. This design is a trade-off between ease of production of the magnet track and a perfect sinusoidal magnetic field. A sinusoidal field is desired as it allows the calculations for the control of the actuator to be quite easy. A perfect sinusoidal magnetic field can be created by an ideal Halbach array, after K. Halbach, who succeeded in producing a sinusoidal magnetic field by a magnet array in which the magnetization vector rotates continuously, as seen in figure 2.3. This array is difficult and expensive to produce and thus is only approximated in most realizations with a magnetization vector that rotates in finite steps. If we put a piece of current carrying wire in this sinusoidal field, there will be a force exerted on it, as depicted in figure 2.3. In ILSMs instead of a straight piece of wire, there is a coil in the approximated sinusoidal magnetic field. As can be seen in figure 2.2, the coil has parts perpendicular and parallel to the magnetic field. Thus forces on a coil in sinusoidal magnetic field can be shown as in figure 2.4. The distance between 2.1 Electro-mechanical a north and a south principles pole of of a Halbach linear synchronous array is defined as the magnet pitch, depicted motors by τ, and can be seen in figure 2.4. The basic principle behind the linear motor as well as the planar motor is the Lorentz force which is generated when a charged particle or point moves through a magnetic field. A comprehensive treatment of electromagnetic force equations can be found in Molenaar [27] chapter 2 and appendix A. The general formulation of the Lorentz force is given in equation 2.1, where F L denotes the Lorentz force, Q the pointcharge, v the velocity and B the local magnetic flux density. F L = Qv B (2.1) The Lorentz force is perpendicular to the direction of the moving particle and the magnetic flux density. The Lorentz force equation is only valid if the surrounding media have a permeability equal or nearly equal to vacuum. Air and rare earth permanent magnets that can hardly be (de-)magnetized with an external field fulfill this require- Figure 2.3: Left: ment [27]. A sinusoidal magnetic field produced by the ideal Halbach array. Right: Forces on In figure a current 2.2 the resulting carrying Lorentz wire force placed of a current perpendicular carrying wire through to a sinusoidal a magnetic magnetic field is shown. Two coils A and B are shown above a permanent magnet array (PMA). field. (figures from [5]) Coil A generates a net force in the vertical direction z and coil B generates a net force in the horizontal direction x, τ is the magnet pitch, the distance between a north and a south pole and which is thus half of the magnetic flux periodicity. Figure 2.2: Force generation through a coil, obtained from Compter [6]. Figure 2.4: Forces on a coil in an approximately sinusoidal magnetic field produced by a segmented Halbach array. (figure from [5]) The electro-mechanic equations of such a coil can be derived in the following way [6]. Note that although the coils have finite length in the y direction end-effects are neglected. The basic electrical equation is the following: u = ir + dϕ ϕm(x, z) dz ϕm(x, z) dx = ir + Ldi + + (2.2) dt dt z dt x dt In this equation u represents the voltage, i the current, R the coil resistance, L the self-inductance and ϕ the coupled flux generated by the magnets in the coil. The self-inductance is assumed to be independent of the z-position. The power equation is obtained by multiplying equation 2.2 with the current which gives: Let us go deeper into the electro-mechanical principles of the ILSM, starting with the Lorentz force equation. Equations in this section are from [6]. Force on a point charge q moving with velocity v or a straight wire of length L in magnetic field with flux B is given as in equations 2.2 and 2.3. If the wire is not straight P = i 2 R dli2 ϕm(x, z) + i dt z dz dt 10 + i ϕm(x, z) x dx dt (2.3)

14 than equation 2.3 can be written instead as equation 2.3, where dl represents the infinitesimal wire piece at location dl. F = q (v B) (2.1) F = I (L B) (2.2) F = I dl B(l ) (2.3) We immediately see that the force on the wire is mutually perpendicular to the magnetic flux and to the wire itself. That is in the composition of figure 2.4, the force on the coil has components in x and z directions. The force constants of a coil in the sinusoidal magnetic field is given by the following equations. K x = F x i K z = F z i = ϕ m x = ϕ m z (2.4) (2.5) Where F x and F z are respectively the forces on the coil in x and z directions, i is the current through the coil and ϕ m id the coupled flux. The coupled flux can be described as follows. ϕ m (x, z) = ϕˆ m e αz/τ sin ( πx ) τ (2.6) Where ϕˆ m is the top value of the flux linkage, α is a geometry determined constant. Combining equations from 2.5 to 2.6 gives the force constants of a single coil as follows. K x = πϕˆ ( m πx ) e αz/τ cos = K ˆ ( πx ) x cos (2.7) τ K z = πϕˆ m e αz/τ cos τ τ ( πx τ ) = ˆK z sin τ ( πx τ ) (2.8) In equations 2.8and 2.8, Kx ˆ and ˆK z denote the maximum values of the corresponding force constants. They are independent of x, hence are constant as the coil moves in x-direction. Finally forces on the coil can be written as follows. F x = ik ˆ ( πx ) x cos (2.9) τ F z = i ˆK ( πx ) z sin (2.10) τ Now that forces on a single coil are established, we can look at the forces on a forcer. A forcer consists of three coils, placed with a distance of 4τ/3 to each other. Thus force constants for coils j = 1, 2, 3 can be written as follows. ( K x,j = K ˆ πx x cos τ + 4π ) (j 1) (2.11) 3 ( K z,j = ˆK πx z sin τ + 4π ) (j 1) (2.12) 3 If the current through each coil is supplied as ( πx i j = Î sin τ + 4π ) (j 1) + φ 3 (2.13) 11

15 Where Î is the amplitude of the current, then the forces on a forcer can be written as ] [ ] Kx,1 K = x,2 K x,3 F z K z,1 K z,2 K z,3 [ Fx i 1 i 2 i 3 = 3 2Î [ ˆ Kx sin (φ) ˆK z cos (φ) ] (2.14) The reverse equation of attaining current and phase using the required forces then becomes as follows. Î (t) = 2 (Fx ) 2 ( ) 2 (t) Fz (t) + (2.15) 3 Kˆ x ˆK z ( ) F x (t) ˆKz φ (t) = arctan (2.16) Kˆ x F z (t) 2.3 Attaining 2-D Forces In this section the mathematics behind the realization of the N-Forcer will be analyzed. Let us look at the idea behind the N-forcer once more, in detail this time. As mentioned before, Apptech came up with slight modifications on the ILSM, that enables these 1-D actuators to produce 2-D forces. The first step of this modification is to change the position of the forcers with respect to the magnet tracks as seen in figure 2.5. Figure 2.5: The repositioning of the forcers in the magnet tracks eliminates the cancellation of the forces produced by horizontal sections of the coils. This allows the modified ILSM to produce two orthogonal forces. (figures from [4]) With this new position of the forcer coils, the forces generated by the horizontal sections of the coils do not cancel out. Thus the system can now produce forces in both x and z directions instead of just x. The second step is to control these forces in z direction. This is done by adding the second commutation. That is instead of trying to control the current through the vertical sections of the coils only, we control the current through the horizontal sections as well. This sentence may seem contradictory as current passing through a coil is the same in everywhere along it, however what we mean with control with respect to sections is the following. Depending on the position of a certain coil with respect to the magnetic field, either horizontal or vertical section of the coil is interacting with the magnetic field, as can be seen in figure 2.6. Thus we control the current in related section of the coil, called x and z commutations, such that the desired forces in x and z directions are 12

16 attained. This enables the generation of two independently controllable orthogonal forces in x and z directions. Figure 2.6: The addition of the second commutation allows us to control the forces produced by the horizontal sections of the coils. This allows the modified ILSM to produce two independently controllable orthogonal forces. (figures from [4]) Now let us see some experimental results that would help the understanding of some parameters that affect the x and z commutations. These results also show the robustness of this novel design. 2.4 Forces on the Forcer Coils: Some Experimental Results I would like to start this section by mentioning that all the following information, figures, graphs, etc. are taken from [4], unless stated otherwise. As mentioned in previous sections, N-Forcer produces two independently controllable orthogonal forces using a modified ILSM. This modification involves two steps, changing the forcer position and adding z-commutation. First, let us analyze the effect of forcer position on forces in x and z directions. The following figures (figure 2.7) show the variation of forces in all directions, i.e. x, y and z directions, as the forcer is moved along z direction in 1 mm steps. First graph in the figure shows how force in x direction (F x ), attained by x commutation, changes when forcer moves in z direction. Similarly, second graph shows how F z, attained by z commutation, changes with forcer position. As can be seen from these graphs, repositioning the forcer in z directions allows us to produce F z, a force in z direction, when z commutation is used. The price of this action on the other hand is a loss in the magnitude of F x, as now some part of the vertical section of the coil is outside the magnetic field. The effect of the first step in the modification is now clear with experimental results. Now let us analyze the effect of imperfect positioning of the forcer between the the magnets, i.e. the effect of y position of the coils. In the following graphs, in figure 2.8 we can see that F x and F z remain unchanged with changing y position for x and z commutations respectively. As a matter of fact, all forces are independent of y position in x commutation. However in z commutation, we see a linearly changing F y. This can be explained physically. Previously, we commented on the fact that the magnetic field is not perfectly sinusoidal. This results in a non zero magnetic field between the magnets which also has an x component, as seen in figure 2.9. This nonzero magnetic filed in x direction is not a problem for x commutation since as seen in figure 2.6, in this position the coil section exposed to this field is parallel to the field direction. However in z commutation, the coil section exposed to the field in x direction lies along z direction. Hence this situation produces a nonzero force in y direction. 13

17 Figure 2.7: The first step in modifying the ILSM to attain 2-D forces is to move the forcer along z direction. This results in a decreased force in x direction and increased force in y direction. (figures from [4]) Figure 2.8: The effect an offset in the position of the forcer along y axis, on the forces produced. (figures from [4]) 2.5 Properties of Actuators Used in the N-Forcer Setup There are more than one ILSM on the N-Forcer setup. Furthermore, there are more than one forcers per ILSM, with a total of N forcer for the setup, hence the name, N-Forcer. In particular there are 4 magnet tracks and 6 forcers. Magnet tracks can be recognized as the red blocks in figure 2.1, attached to the frame. Four of the forcers can be recognized as yellow plates attached to the chuck in the same figure. The remaining two are the orange plates attached to the chuck. Let us refer to the upper magnet tracks as z tracks and the lower magnet tracks as y tracks. From this figure one can easily deduce that the 4 yellow forcers move along z tracks and 2 orange forcers move along y tracks. Yellow forcers with z tracks produce 4 forces in x and 4 forces z direction, as there are 4 forcers. Orange forcers with y tracks produce 2 forces in x and 2 forces in y direction. Two different ILSM are used for the z tracks and the y tracks. Z tracks use UM series ILSM and y tracks use UC series ILSM. The data related to these products can be found in appendix B.1. 14

18 Figure 2.9: The imperfections in the magnetic field, depicted with red arrows, cause a nonzero force in y direction. (figure from [4]) 2.6 Measurement System Position measurements of the N-Forcer are carried out with laser interferometry. Laser interferometry measures the position of the chuck in all 6 dof. So there are measurements for x, y, z translational directions and Rx, Ry, Rz rotational directions. The points on the chuck at which these measurements are taken are neither its center of gravity nor the axis of rotation. Thus, measured quantities are not completely decoupled and some measures need to be taken while analyzing these data. In addition to this care should be taken to avoid any geometrical errors during measurement by taking precautions such as bringing the machine to constant temperature. Although laser interferometry would be supported by a separate frame that is isolated from ground vibrations in a conventional setup, in the N-Forcer, it is mounted on the same frame as the chuck. This is because the N-Forcer is only a trial setup for now and separating the measurement frame would have added to its cost. In addition to positions, the currents delivered to the forcers are also recorded and used to calculate the forces on the chuck. 2.7 The Controller Currently a feedback controller and a feedforward controller, feeding only mass, are incorporated in the system. N-Forcer is currently using a feedback controller and a feedforward controller. The bode plot for the feedback controller can be seen in figure The open loop and sensitivity Bode plots are also present in figure 2.11, respectively. As can be seen from these plots, system has a bandwidth of 83.5 Hz. The feedforward controller is composed of acceleration feedforward and a position dependent force compensation table. Mass used in the acceleration feedforward is 5.6, while the actual mass of the chuck is around 3 kg. The difference in between results from some gains in between while implementing the feedforward controller. The position dependent force compensation table compensates for the forces that are there due to the physical imperfections of the system like unequal magnet powers. This table is built up by measuring the force applied on the chuck by the feedback controller in a slow speed scan of whole track. The feedforward controller is further analyzed and changed in the following chapters. Detailed analysis can be found in chapter 4, sections 4.2 and 4.3. Now that the main function of the N-Forcer, its components, and the novelty of the idea behind its development are clear, we can proceed with a more detailed analysis. In the coming chapters, the components, actuation principles, control methods will be discussed. 15

19 Figure 2.10: Bode plot of the feedback controller of the N-Forcer. 16

20 Figure 2.11: Open loop and sensitivity bode plots of the N-Forcer with the feedback controller. 17

21 Chapter 3 K-factor Compensation 3.1 K-Factor Compensation: Aim Magnetic actuators show position and current dependent force ripples mainly due to imperfect magnetic fields produced by the magnet arrays. It is also common for systems to have other position dependent, repeated disturbances arising from cable forces, cogging forces, etc. As explained in chapter??, forces produced by synchronous motors can be modeled as the multiplication of a position dependent force constant and the current passing through the coils in the case of perfect sinusoidal magnetic field. However magnet arrays do not produce perfect sinusoidal fields and thus some compensation is necessary. The real force produced by the motor as opposed to the desired force can be represented as follows. F real = K (x) F desired + F disturbance (3.1) The nature of the disturbance forces are position dependent and periodic. Thus K-factor compensation is proposed to eliminate these disturbances. Jan-Maarten Prevoo from TU Delft worked on implementing a kind of iterative learning control for non-repetitive motion using K-factor compensation. This chapter look into theoretical background of his work. 3.2 K-Factor in the ILSM To test for K-Factor related errors, other large errors in the system should be compensated first which makes up most of this report. After this step, a power spectrum analysis on the error signal will show if there are any periodic components. Of course periodicity in time domain should first be converted into position domain by using the velocity data. A study on ILSM shows, when period spectrum of the controller output is analayzed for unloaded chuck moving at constant speed, magnitude of the discrete Fourier coefficients peaks at the periodicity of the magnet track. This is displayed in the following figure. On the other hand, for loaded chuck moving at constant speed, coefficients peak at other harmonics of the track periodicty as well, as depicted in the next figure. 3.3 K-Factor Compensation Algorithm The suggested algorithm for the compensation of the error displayed in the previous section, an iterative learning feedforward controller making use of position and current data is proposed. The periodicity of the magnets in the magnet tracks are of 18

22 Figure 3.1: The imperfections in the magnetic field cause disturbances in ILSM with the period of the magnet track. Figure 3.2: ILSM with a loaded chuck show error signal peaks at the track period and its harmonics importance in this algorithm as errors are expected in its harmonics. Details of the implemented algorithm can be found in J. M. Prevoo s work, in [1]. The learning algorithm involves an optimization with the goal of minimization of the tracking error. The aim of control law developed here is to compensate for the K-factor variation, which is a position dependent gain which is only distinguishable when actuators apply force on the system. Thus the learning should be performed during accelerating profiles. In addition to this, for the learning algorithm to compensate only the K-factor related errors, other position dependent constant disturbances should be completely compensated for. The compensation signal is approximated by sine based or pulse based functions. 19

23 The cause of this hysteresis effect is not further investigated, instead it is decided to only compensate for the forward motion which is most reproducible. The learning algorithm is adjusted such that it will only learn and compensate for the forward motion profiles Compensation To compensate for these position dependent constant force disturbances use is made of a simple cogging table. The values in this table are obtained using a low constant velocity scan and capturing the position and force values. Because the N-forcer is not accelerating, the force acting on the plant F real is zero, see equation 1.5, and the force out of the controller F des is thus the force needed to compensate for the position dependent disturbance, equation 3.2 thus becomes: ˆF dist (x) = F dist(x) K(x) (3.4) Where ˆF dist denotes the compensation of the position dependent disturbance. Implementing this in the feedforward path results in a transfer of one. Since the position dependent K-factor is also in this disturbance compensation, it is expected that amplifier offset as well as amplifier gain effects will be visible in the ˆF dist signal. K ff ˆK ( x, θ) ˆF dist (x) F dist (x) Plant u ff r e K fb u kfb u K(x) P x Figure 3.15: Plant with complete compensation Figure 3.3: Plant with complete compensation, figure from [1] To prevent that the acceleration phase is part of the table, the stroke for creating the table is made 5 mm. longer on both sides then the working area of the motion profiles. The motion profiles are executed over a range of 60 mm. whereas the compensation table has a range of 70 mm. Because there is a limit to the number of data points the lowest speed that could be executed was 2.8 mm/s. With a sampling frequency of

24 Chapter 4 Experimental Work On the N-Forcer Setup The analysis on the N-Forcer introduced the need of doing some experiments either to check the behavior of the system or to calibrate certain parameters. I performed many experiments on the setup and analyzed the data. This chapter presents the experiments I have done with the reason for doing them, the outcome and the resulting correction on the setup, if there are any. Some results of these experiments led to improvement of the controller. Explanation about the initial composition of the controller may found in section Directionality of Position Dependent Disturbance Force Compensation Table As previously explained in section 2.7, N-Forcer utilizes a force compensation table. This table compensates for position dependent disturbance forces, mainly resulting from the inaccuracies of the magnetic field produced by the magnet tracks. In the initial controller, force compensation table is calculated as follows. The chuck travels from zero position of the track to the maximum possible position with a very slow velocity (about 4-6 mm/s) and turns back. The output of the feedback controller, i.e. the force going to the chuck, is recorded for both directions. Then it is averaged over these two measurements, is tabulated with respect to the position data and is incorporated into the controller. It is obvious that this method would work intuitively but we can also conclude it by looking at the Bode plot of the complementary sensitivity function, u/f x = OL/ (1 + OL) (see figure 4.5), as seen below at figure 4.1. Since the gain of this function is 0 db, i.e. 1, until 10 Hz, and since more than 99.9% of the power of Fx signal is contained up to 4 Hz at these slow velocities (see figure 4.2) we can judge that u = F x, i.e. output of the feedback controller is equal to the disturbance forces that act on the plant. The reason for averaging the forces measured in forward and backward scans is the fact that these two measurements have an almost constant difference between them as seen in figure 4.3. This difference is about 0.5 N and is considerable as the disturbance forces we are trying to compensate for are about the same order of magnitude. Thus the source of this difference should be found and proper action should be taken. My guess on the source of the constant difference is the unleveled granite table. The measurement coordinate of the system is defined with respect to this table. Thus the forces measure in x-axis are the forces that are parallel to the surface of 21

25 Figure 4.1: The gain of the signal u/f x is 0 db until 10 Hz. Figure 4.2: Most of the power of the controller output signal is contained up to 10 Hz. 22

26 Figure 4.3: There is a clear difference between the disturbance forces on the system in forward (shown with F in the legend) and backward scans (B). This difference is almost constant over position. (figure from J.M. Prevoo) the table. Since no measure is taken to align the surface of the perpendicular to the gravitational acceleration direction, it might be that table is positioned slightly misaligned and thus x force has a component that is doing work against gravity in one direction while trying to carry the mass up and the reverse in the other direction. To test this I thought of an experiment where the table is further tilted toward one side and forces are measured like this. As a result of this experiment, we saw that offset was further increased by the inclination of the granite table. Since we found out that the table was tilted, we decided to compensate the forces only with the forward scan and do the other experiments while chuck moves forward only. 4.2 Tuning of the Feedforward Controller In this section, acceleration feedforward and delay tuning of the feedforward controller will be analyzed. The work on the N-Forcer shows that the error during standstill is about 60 nm with the initial controller. This error is composed of a 50 Hz oscillation around 0 and does not have any other pattern in it. Since the oscillation is exactly at 50 Hz, the cause is thought to be electrical as the electricity from the mains is 50 Hz. This error can be overcome by analyzing the electrical components in the system and improving them. However error during move is much larger, and its pattern can be explained in relation to the properties of the motion that produce that error. The first kind of motion to be analyzed is accelerating move. It is obvious that if no feedforward is used, there will be large errors during acceleration as feedback controller can only reduce an existing error and can not prevent it beforehand. However a feedforward controller can supply the force needed for the desired acceleration, thus prevents large errors before they occur. N-Forcer already has an acceleration feedforward with mass coefficient 5.6. The error for an acceleratingdecelerating motion profile with this feedforward is shown in figure 4.4. As seen from figure 4.4, the error in the accelerating regime is not well com- 23

27 Figure 4.4: The acceleration feedforward with mass coefficient 5.6, fails to compensate for the error during acceleration. The error is shown in blue and the scaled feedforward force in green. Sign ẏ positive positive ÿ positive positive e = y ref y positive negative mass coefficient low high Table 4.1: Evaluating mass coefficient for various values of error. pensated, thus the feedforward controller needs further tuning. When we analyze the error, we see that during positive velocity and acceleration, first it is small and positive then it becomes large and negative. If it was only positive in positive velocity and acceleration regime we would explain this by a lower than necessary mass coefficient. On the contrary if it was negative, we would explain this by a mass coefficient larger than necessary. However the observed behavior can not be explained by pure acceleration feedforward. The suggested explanation for this error behavior of the system is that the acceleration feedforward is applied too late, causing an initial positive error but the mass coefficient is too large, causing a negative error for the rest of the regime. Then we should compensate for delay as well. The suggested way to do this is represented in the following block diagram in figure 4.5. The delaying algorithm is Sign ẏ positive positive ÿ positive positive e = y ref y positive negative delay = (time instant at which pos. ref. is inserted) too little too much - (time instant at which ff signal is inserted) (ff too late) (ff too soon) Table 4.2: Evaluating delay for various values of error. 24

28 explained by a simpler example in appendix A.3.2. Figure 4.5: The required compensation for the delay is not necessarily an integer multiple of the sampling time however only delays equal to the integer multiples of sampling time can be inserted into a real system. The way to overcome this is to implement delays as seen in this model This model delays to position reference by 1.3 times the sampling time with respect to the feedforward signal. Since there are two parameters that effect the error we are analyzing, the tuning procedure will be an iterative one and will start with delay tuning. As explained previously, it seems that the position reference signal is given too early with respect to the acceleration feedforward signal. Or if we state this in a causal manner, acceleration feedforward should be supplied earlier as it takes time to effect the system. This is because the real plant can be modeled as a mass and a time delay. We can measure the amount of this delay from frequency response measurements of the plant by plotting the phase of the plant against frequency on a linear scale. Further explanation on this matter can be found in appendix A.3.1. Below in figures 4.6 and 4.7 the Bode plot of the plant and its phase against frequency are depicted. The data to plot these are taken from a frequency response measurement. The calculated delay using three different measurements on the plant, performed by J.M. Prevoo and D. Denie, came out as 268µs from two measurements and 396µs from the third. Thus the position reference is delayed by an amount of by two sample times, 400µs with respect to the acceleration feedforward as an initial guess, as its implementation is easier. The result is shown in figure 4.8, with the red plot. This plot shows that the delay tuning improved the response. Thus I start tuning the other parameter, the mass coefficient. As stated previously, the initial mass coefficient of the feedforward controller is higher than it should be. Thus I try a lower coefficient of 5.4 as an initial guess. The result of this modification is seen in figure 4.8, with the purple plot. While this plot shows that the modification improved the response a lot, there still is a systematic error, either because the mass coefficient is too high or because the applied delay is too much. Thus I go on with another iteration where I try the values for mass and delay shown in table 4.3. The results of the trials shown in table 4.3 are shown in figure 4.9. This plot suggests that the best values for mass and delay among the tries were 5.4 and 268[µs] respectively, as shown by the light blue plot as data 6. This verifies the previous guess of 400[µs] delay being too much but eliminates the possibility of 5.4 as mass coefficient being too high. However, this blue plot still contains some systematic error that can be further eliminated. Thus I go on with one more experiment. The light blue plot shown in figure 4.9 in combination with the results stated above suggests that delaying the position reference by 268µs is too much as error 25

29 Figure 4.6: The Bode plot of the plant attained by a frequency response measurement of the system. Figure 4.7: Phase of the plant plotted against frequency on a linear scale. The green line shows the fitted polynomial by the m-file presented at appendix A.3.1. mass Exp # delay [µs] Exp # delay [µs] Table 4.3: Values of mass and delay for first iteration 26

30 Figure 4.8: Response of the plant with different feedforwards. Blue plot shows the acceleration profile, for rest of the plots (mass, delay [µs]): green (5.6, 0); red (5.6, 400); purple (5.4, 400) Figure 4.9: Error with different feedforward signals shown in table

31 mass Exp # delay [µs] Exp # delay [µs] Exp # delay [µs] Table 4.4: Values of mass and delay for second iteration Figure 4.10: Error with different feedforward signals shown in table 4.4. becomes slightly negative in the beginning and the mass coefficient 5.4 is too little as the error becomes positive afterward. According to this I try new values for the control parameters that are shown in table 4.4. Although I have a guess on which direction to change the parameters, I try both ways to validate my guess. I expect the values represented by the data number 9 to give the best result. The result of the experiment for the trial of the values shown in table 4.4 is given in figure This figure shows that decreasing delay and increasing mass improves the response as expected but only in the first part of the motion profile. On the other hand it makes it worse on the second part. Thus it is difficult to decide which values of the parameters is the best. To decide on which value is actually the best, I checked the the first norms of the errors for all 9 data for 20 different motion profiles (these profiles are by J.M. Prevoo). This norm gives the absolute values of the error summed over each sample. The m-file used for this purpose can be found in appendix A.4. The result suggests that 268µs is the best value for delay correction for most of the profiles. The result is a bit blurred for mass coefficient and 5.40 are best for 6 profiles each and 5.41 is best for remaining 12 profiles. However, some profiles have other disturbances that can effect the judgment of mass coefficient due to their high accelerations and jerks. Thus I chose one profile which I think is most disturbance free, which is shown in figures throughout this section, to analyze the control parameters. Fort this profile, 5.4 seems to be the best as mass coefficient, shown by the blue dotted plot in figure 4.10, representing the data set 8. As these values of the control parameters are the same with the previous iteration, tuning of the control parameters delay and acceleration feedforward is finalized. 28

32 Figure 4.11: Error at constant velocity increases as velocity increases. 4.3 Robustness of Position Dependent Force Compensation Table Against Velocity As stated previously, feedforward controller of the N-Forcer has a position dependent disturbance force compensation table in it. This table is built by measuring the disturbance forces while chuck scans the track at a slow velocity as explained in detail at section 2.7. This table is supposed to eliminate the errors completely for constant velocity motion as there are no suspected disturbance forces other than the position dependent ones and acceleration related ones. However when the velocity is high, there exists an error pattern for constant velocity motion as can be seen in figure The reason for this seeming velocity dependence may show existence of some velocity dependent errors. This can be tested by reconstructing the disturbance forces. To reconstruct disturbance forces one should implement an inverse process sensitivity and plant filters and subtract one signal from the other as can be seen in figure The inversion can be done using ZPETC (more detail on this algorithm is in [7]). The result of inversion can be seen in figure??. These disturbance forces are plotted in figure As can be seen in this figure, there is no velocity dependence in disturbance forces. Then it is concluded that the increase observed in the error signal is due to the fact that the frequency of error signal increases with speed where effectiveness of the feedback controller becomes less. 4.4 Difference of the Error Pattern of the First Scans As explained previously in section 4.1, experiments on the N-Forcer are conducted on single direction only. The chuck is moved in the opposite direction only to for repositioning. In the experiments where same motion profile is executed several times, a difference in the error pattern of the first scan is observed, as can be seen in figure This difference is also visible in the outputs of the feedback controller. 29

33 Figure 4.12: Reconstructing disturbance forces using inverse filters. Inversion can be done using ZPETC algorithms. Figure 4.13: ZPETC gives good approximation of inverse plant and process sensitivity filters with finite magnitude gains. 30

34 Figure 4.14: Reconstructed disturbance forces do not show any velocity dependence. Figure 4.15: In a single experiment, chuck scans the track of 6cm three times. Blue plot shows the first scan, green second scan and red third scan. While green and red plot seem to resemble each other more, blue plot differs from them, especially in the first bits. Scan speed is 20mm/s 31

35 Figure 4.16: When the table allowed to come to standstill between scans, every error signal is different as the first scan case. We suspect the rocking of the granite table for this difference. The first scan starts while table is standing still. However other scans start while table is rocking due to the input from high accelerations in repositioning scan. To test if this difference in initial conditions cause the difference of the error pattern of the first scan, I decided to do an experiment where the chuck scans the track many times during the same experiment. I divided the scans into two groups. In the first group, waiting time between repositioning scan and next forward scan is small as in usual experiments, so I expect to see the first scan difference. In the second group, this difference is more, letting the granite table to come to standstill before the next forward scan starts. The result of this experiment is seen in figure Analyzing the MIMO System The N-Forcer is a MIMO system and it is not perfectly decoupled. Forces applied in a certain direction creates moments in other directions and also the measurement points are not aligned with the center of gravity of the motion system. Rotations in ry, rz directions cause displacements in x. In addition to this, position dependent disturbances exist not only in x-direction but in other direction as well. Thus it is proposed to implement position dependent disturbance force compensation tables in ry and rz directions just like it was done for x direction. This results in a large improvements as can be seen in the following figures. 32

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