Control of a High Speed Machining Spindle via µ-synthesis

Transcription

1 Control of a High Speed Machining Spindle via µ-synthesis Carl R. Knospe Roger L. Fittro Assistant Professor Research Assistant University of Virginia, School of Engineering & Applied Sciences, Department of Mechanical, Aerospace and Nuclear Engineering, Charlottesville, Virginia, Tel : (804) ; crk4y@virginia.edu L. Scott Stephens, Assistant Professor 2508 CEBA, Louisiana State University, Department of Mechanical Engineering, Baton Rouge, LA Tel : (504) ; Fax : (504) ; stephens@me.lsu.edu Abstract: Due to high surface speed and active control capabilities, active magnetic bearings hold great promise for high speed machining spindles. Herein, the control problem posed by this application is examined and the development of an advanced prototype is reviewed. A µ - synthesis framework is proposed for this problem and it is shown that the minimization of the susceptibility to machining chatter may be easily put into this framework. In addition to handling uncertainties in sensor and actuator components, this formulation may also include an uncertainty representing the range of cutting tools for the spindle. As the machining spindle is still undergoing preliminary testing, experimental results are presented for another magnetically suspended rotor which strongly indicate the effectiveness of the proposed design procedure. 1 Introduction Active magnetic bearings (AMB) have several advantages over conventional rolling element bearings for high speed machining. First, the maximum achievable surface speed of magnetic bearings is much greater than rolling element bearings. While the surface speed of ceramic rolling element bearings has been limited to about 2 million DN (shaft diameter in mm multiplied by shaft speed in RPM), the surface speed of magnetic bearings can reach as high as 5.0 million DN. The higher attainable surface speed permits spindles with greater diameter and, therefore, greater stiffness. Second, because they are non-contacting, magnetic bearings are not prone to the same thermal problems as conventional bearings at high speed, such as thermal lock up. Last, the active nature of magnetic bearings can be exploited to achieve much lower vibration levels than obtainable with rolling element bearings through increased damping, coordinated action, and adaptive "balancing" capabilities. Over the last three years, the authors have been developing a new high speed machining spindle controlled by active magnetic bearings. This spindle has recently been magnetically levitated via feedback control and is undergoing preliminary testing. In this paper, we will review this application, discuss a control design method, and present experimental results obtained from another magnetically supported rotor. Active Magnetic Bearings Figure 1 illustrates the components of an active magnetic bearing. The magnetic actuator is composed of attractive electromagnets that are arranged around a magnetically permeable journal attached to the shaft. Position sensors, usually placed along the shaft next to the actuator, measure the x and y displacements of the shaft. These measurements are used by a feedback control system to determine the currents which are applied to the actuator's coils so as to suspend the shaft. In response to a displacement, a simple control system would increase the coil current on those poles opposite the displacement and decrease the current on those poles in the direction of the displacement. This action produces a restoring force which tends to stabilize the system. Typically, the current in the coils is operated about a bias level so as to have a significant force slew capability when there is no net force. This causes the plant to be open loop unstable. Figure 1: An active magnetic bearing 2 High Speed Spindle Several magnetic bearing machining spindles are presently in production or laboratory testing [1-4]. Experience of some industrial users with commercially available magnetic

2 bearing spindles suggest that the full capability of these devices has not been achieved. To further explore their potential, the authors have been developing a high speed magnetic bearing spindle and researching the control challenges presented by it. The completed spindle, now undergoing preliminary testing at the University of Virginia, has been designed with a significant increase in available power (>100%) and allowable cutting force (>100%) over similar commercial spindles. The magnetic bearing spindle uses differential optical sensors to determine the radial and axial position of the shaft. Optical sensors have the advantage of being immune to magnetic fields and therefore have very low noise. This is an important characteristic for the high gain feedback control which the bearings will use. The sensors, designed at the University of Virginia, are based on the occlusion of a light beam from an infrared LED to a phototransistor by the shaft. Experimental results indicate that these sensors have an rms noise level less than 200 nm (8 µ inch). The magnetic bearing spindle is controlled by a parallel processing digital controller designed and built at the University of Virginia [5]. Four Texas Instruments TMS320C40 digital signal processors are employed for control algorithm computations. Controller throughput will depend on algorithm execution times; preliminary testing indicates a throughput rate in excess of 10 KHz. Figure 2 shows a schematic of the AMB machining spindle with three radial magnetic bearings, termed the nose, midspan, and tail bearings. The shaft is dual level with a large rotor on which all bearing journals and the motor rotor are carried, and a smaller drawbar located within an internal bore of the main rotor. The purpose of the drawbar is to actuate the cutting tool, allowing for tool changes, as several different tools may be used during a machining operation. The cutting force applied at the tool tip is also shown. The objective is to control the flexible shaft such that motion at the cutting tool tip is minimized. 3 Control Theory Previous Work Today, almost all magnetic bearing machining spindles use simple decentralized feedback control, often proportionalintegral-derivative (PID) control. This type of algorithm feeds the displacement signal from each spindle position sensor back to only the closest actuator. With this method, each magnetic bearing acts much like a conventional bearing, being characterized by a stiffness and a damping. In some cases, additional filtering is added in the feedback loop to decrease the effective stiffness and damping at the operating speed so as to cause the shaft to rotate about its inertial axis. Several researchers have investigated more advanced control algorithms such as H and µ-synthesis for a variety of magnetic bearing applications. As argued below, these methods seems well suited to the control of machining spindles. In 1994, Nonami and Ito [6] first experimentally demonstrated µ synthesis for support of a rigid rotor in magnetic bearings. They employed a centralized 5-axis controller that was designed using only the shaft s rigid body modes in the model. Earlier, Fujita [7] presented experimental results for µ-synthesized controllers suspending a non-rotating beam in magnetic bearings. In both works, experimental results showed superior performance of the µ-synthesized controllers over H loop shaping and PID controllers. In both cases, the performance considered was low frequency output disturbance rejection in the presence of essentially unstructured input multiplicative or additive uncertainty. Due to the conservative nature of the uncertainty representation, a significant amount of performance was sacrificed in this problem formulation. µ-synthesis Since the cutting forces vary drastically over the range of spindle speeds, feedrates, cutting tools, and cutting conditions (e.g., slotting, face milling, end milling), and the quality of the cut is directly dependent on tool vibration, the control problem posed has an H performance requirement. Plant uncertainties must be included in the design procedure since the rotor is very lightly damped. Fortunately, the uncertainties in this system are highly structured (often parametric). Therefore, controllers may be designed which sacrifice only a little performance to achieve the necessary robustness. The µ-synthesis procedure permits the design of multivariable controllers for complex, uncertain linear systems so as to minimize an H cost function. It is a rather natural augmentation of H control theory with the analysis techniques of the structured singular value µ [8] Figure 2: AMB Machining Spindle The essential principle behind µ-synthesis is the modeling of the uncertain plant as a linear fractional transformation

3 (LFT) of a nominal system model and an uncertainty block s, which may have a structure. These uncertainties are ( ) stable and norm bounded such that ( s ) < 1. (The norm bound condition may always be achieved since stable, minimum phase, frequency-dependent weighting functions W s may be appended to the nominal plant model. The ( ) LFT representation adds pseudo-inputs and pseudo-outputs to the nominal plant to form an augmented plant G(s). The goal of the control design is to minimize the dynamicallyscaled H norm from the disturbance and pseudo- inputs to the performance and pseudo- outputs. For dynamic uncertainties (as opposed to real parametric) this may be written as K 1 ( ) F ( ( ) ( )) ( ) min D s l G s, K s D s (1) ( s), D( s) where F l indicates the lower LFT and the dynamic scales D(s) must be chosen with a structure complimentary to that s. The design problem posed by Eqn. (1), known as of ( ) µ synthesis, is usually solved by the DK iteration method. This procedure minimizes the cost function by iterating between minimizing the scaled H norm with respect to D(s) and with respect to K(s). Both of these problems are convex and may be easily solved. However, since the problem is not jointly convex, the solution so obtained will usually not be optimal. One consequence of this, encountered by the authors, is that increasing the number of actuators and sensors available to the controller may decreased performance achieved by DK iteration. 4 Model, Uncertainty and Performance Model and Uncertainty Representation To model the AMB spindle, models of the magnetic actuators, rotor, amplifiers, sensors, anti-aliasing filters, and digital controller must be obtained. Also, uncertainty descriptions are needed. For the spindle under consideration, a planar, dual level model of the rotor and drawbar may be used since the gyroscopic effects are negligible. A significant uncertainty is introduced into the spindle rotordynamics because of the variability in the tool mass arising from the large number of tools that will be used. This results in a 15% variation in the rotor s first flexible-mode natural frequency. Our experience is that the rotor s dynamics may be characterized very accurately if the cutting tool is fixed. Therefore, no other uncertainty is needed for this component. Four flexible modes of the rotor-drawbar are within the target controller bandwidth of 2 KHz. These modes may be assumed to have a modal damping of about 1%. The optical sensors have very high bandwidth and may be modeled as a gain. The power amplifiers may be modeled fairly accurately using a first order low-pass filter with a bandwidth greater than 1 KHz. Performance The cutting force depends upon many factors which are specific to the machine tool and type of cut. Description of such factors is beyond the scope of this paper. The cutting force will be predominantly composed of harmonics of the cutter tooth-pass frequency (forced vibration) and perhaps a machining chatter frequency (self excited vibration). The controller design goals are to (i) reduce the tool harmonic response at multiples of the tooth-pass frequency, f ct, and, (ii) suppress the onset of machining chatter which occurs at chatter frequency, f c. The first of these goals may be accomplished by minimizing the H norm of the cutting tool dynamic compliance since the spindle speed and number of cutter teeth varies during spindle operation. Machining chatter, which occurs at different combinations of running speed and axial width of cut in end milling, can be characterized by a maximum width of cut which yields stable machining under all conditions. This maximum cut width is related to the tool compliance G c (s) by 1 b (2) Re where Re[Gc] is the real part of the oriented transfer function of the tool. Without specifying a specific cutting condition, chatter should be assumed to occur at the most negative real part of the dynamic compliance of the cutting tool. Therefore, in order to optimize with respect to goal (ii) listed above, the minimum real tool compliance of the spindle should be maximized by the control design. Figure 3 illustrates a typical augmented model for design problem considered. Shown are the nominal system, the uncertainties and associated weighting functions, and the performance weighting functions. Multiplicative input and output uncertainties are used to represent the inaccuracy in the amplifier/actuator and sensor transfer functions. Tool mass variation may be represented as shown by a real parametric uncertainty δ. Since the tool mass is distributed in reality, the lumped mass uncertainty is fed back at two locations along the length of the nominal cutting tool. Referring once again to Figure 3, the performance weights W t, W r and W c which limit direct tool compliance, real tool compliance and controller authority, respectively, are shown. The proper choice of these weighting functions is discussed in [9]. The performance inputs are shown as w=[w r,w t,w c1,w c2,w c3 ] and the performance outputs are shown as v=[v r,v t,v c1,v c2,v c3 ]. Also shown are the bearing G c

4 Figure 3: AMB Machining Spindle Model forces, F n, F m, F t, and the shaft displacements, X n, X m, X t at the nose, midspan and tail sensors respectively. Finally, inputs F c1 and F c2 represent applied forces at the cutting tool tip and tool midspan, and outputs X c1 and X c2 represent the displacement at the cutting tool tip and tool midspan, respectively. As previously shown, the minimum real tool compliance, Re[Gc], is an index of the tendency of the spindle to chatter and therefore should be maximized. The real tool compliance specification may be incorporated into the design problem by introducing a negative unity feedback loop around the tool compliance transfer function G c. as shown in Figure 4. In this case P < 1 implies that Re[ Gc ] > 1 2W r. Thus, the chatter suppression problem may be formulated as an H one. From the authors experience with designing controllers for the high-speed spindle via DK iteration, the following general conclusions are offered: (1) the µ-synthesis framework is well suited to the problem; both absolute and real tool compliance may be minimized; (2) when a controller is designed that is robust to a large variation in cutter mass, very poor performance is achieved; (3) a performance specification which limits shaft deflection at locations along the shaft with small clearances is required since the controller may allow large shaft deflections in order to minimize deflections occurring at the tool. 5 Experimental Results Figure 4: Real Tool Compliance Performance Block Apparatus and Modeling In order to investigate the ability of µ synthesis design for this problem under realistic plant uncertainty, a very flexible demonstration rig was used. (Figure 5) The rotor system is made up of a 12 mm diameter shaft with a span of 508 mm supported at either end by radial magnetic bearings. The goal of the experiment was to minimize the dynamic compliance of the rotor midspan using only the actuators and sensors at the two rotor ends.

5 representation for model uncertainties. The uncertainties in actuator gain and bearing negative stiffness for the two bearings were modeled by four scalar real (non-repeated) blocks representing an uncertainty of ±15% in each parameter. The magnitude and phase error associated with the actuator transfer function approximation was incorporated via a complex scalar block for each of the actuators. The weighting for these blocks was chosen so that the uncertainty represented was negligible below 30 rad/s (5 Hz) and increased quickly afterward to a maximum variation of ± 86.6% in gain or ± 60 in phase at 3000 rad/s (500 Hz). Two real blocks were included to represent uncertainties in the sensor gains of ± 2%. Figure 5: Experimental test rig The radial bearings have solid (non-laminated) stators and a laminated journal. Eddy currents in the bearing stators have a profound effect on system performance; accurately modeling this was a major effort. Adjacent to each radial bearing is an eddy current position sensor used for feedback. While the rig has a midspan sensor, this was not used for feedback control. A 0.82 kg disk is mounted at the midspan, and the rotor is driven through a coupling by a servo motor at one end. A model of the rotor was developed that included the first six free-free modes. The natural frequencies of the first three bending modes are 383, 1835, and 2746 rad/s. Models of the other system components were also developed and compared to experimental data. To further verify our system model, the rig was operated with a number of different PD controllers which effectively varied the system natural frequencies. A good match was found between the model and closed loop experimental data in all of these cases. The remaining discrepancies between the component models and experimental data were taken into account in the µ-synthesis design procedure by utilizing structured uncertainty blocks. Details of modeling are provided in [10]. A benchmark decentralized PD controller was designed so that we could accurately assess the performance improvement obtained with multivariable control. Due to the lack of a design methodology with the capability to include the performance specification, an exhaustive search (employing optimization methods) was performed to find the best performing PD controller. The performance of a candidate design was evaluated by determining the maximum midspan dynamic compliance as measured by the H norm. µ Controller Design The insights gained from our experience with system modeling were employed in determining the proper The point compliance performance specification and a controller maximum gain constraint (including roll-off) were also integrated into the µ-synthesis design procedure via two additional complex blocks. The µ-synthesis procedure produced a controller design which robustly met the performance specifications. A very interesting result is that the synthesized controller is open loop unstable. Since the number of states of the controller was large (56 states), a controller reduction procedure was used to reduce it to 23rd order. The controller still had two unstable poles at ± j518.2 rad/s. The frequency response of the µ controller is shown in Figure 6. Theoretical Comparison The maximum point compliance of the nominal system with the µ controller was 18.8 µm/n, approximately 30% better than that obtained with the optimal PD controller, 26.7 µm/n. The best performance that can be achieved (as determined by H theory) was 12.0 µm/n. This controller is not practical as it has infinite gain and no robustness. However, this does establish a fundamental limit on achievable performance. We also examined (via µ- synthesis) the level of performance that could be achieved for the nominal system when the controller gain was limited in the same manner as in the robust µ controller. In this case, a peak compliance of 14.3 µm/n was achieved. Magnitude (V/V) Frequency (rad/sec) Figure 6: µ controller frequency response

6 This indicates that a significant degree of nominal performance is sacrificed to achieve the desired robustness. Furthermore, it indicates that the important restriction on the performance of our candidate design is the required degree of robustness, not the limitation on controller gain. Thus, to achieve higher performance from our test rig, a more accurate plant model would be of greater benefit than higher controller gain. It is also interesting to compare the above performance figures to the static compliance achieved when the bearings are pinned (obtained with PID control), 10.3 µm/n. Experimental Results The DSP-based digital controller was programmed in assembly language to implement the control algorithm as a discrete time, state space system. Since the µ controller cannot levitate the rotor alone, the program first brings the rotor into support under a low performance, robust PD control algorithm. After 10 seconds, the program switches to the µ control algorithm. Both the µ and PD control algorithms are implemented with a sampling rate of 10 KHz. Figures 7 and 8 shows the experimentally determined and theoretical compliance for the optimal PD and µ controllers respectively. For both controllers, there is a good agreement between theory and experiment (note that the magnitude scale is linear). The peak compliance for the µ controller is 20 µm/n, a 30% improvement over that of the PD controller: 28.5 µm/n. This performance should be compared to that theoretically achievable for the nominal system without any limitation on controller gain, 12.0 µm/n. Clearly, the µ controller offers a significant improvement when one considers what is theoretically possible. The rotor with µ controller has been successfully operated over its entire speed range, 0 to 5000 rpm. 8 Conclusions As the experimental results presented demonstrate, µ- synthesis can be employed to provide very good control of dynamic compliance for rotor systems with structured uncertainty. This capability offers a tremendous potential to the control of high speed machining spindles. References [1] Machine Design, "Magnetic Bearings Holds Spindle For Milling", November 9, 1989, pg. 56 [2] American Machinist, "Magnetically Levitated Spindle to Debut, Delivers Up to 52 kw at 40,000 RPM", August 1989, pp [3] Nonami, K., et. al., "H Control of Milling AMB Spindle", Fourth International Symposium on Magnetic Bearings, August 1994, ETH Zurich, Switzerland, pp [4] Brunet, M., and Wagner, B., "Analysis of the Performance of an AMB Spindle in Creep Feed Grinding", Fourth International Symposium on Magnetic Bearings, August 1994, ETH Zurich, pp [5] Fedigan, S.J., Williams, R.D., Shen, F., and Ross, R.A., Design and Implementation of a Fault Tolerant Magnetic Bearing Controller, Proceedings of the 5th International Symposium on Magnetic Bearings, Kanazawa, Japan, August 28-30, [6] Nonami, K., and Takayuki, I., " µ-synthesis of Flexible Rotor Magnetic Bearing Systems", Fourth International Symposium on Magnetic Bearings, August 1994, ETH, Zurich, Switzerland, pp [7] Fujita, M., et. al., "µ Analysis and Synthesis of a Flexible Beam Magnetic Suspension System", Third International Symposium on Magnetic Bearings, University of Virginia, Charlottesville, VA, 1992, pp [8] Zhou, K., Doyle, J., and Glover, K., Robust and Optimal Control, Prentice Hall, Upper Saddle, NJ, [9] Stephens, L.S. and Knospe, C.R., µ-synthesis Based, Robust Controller Design for AMB Machining Spindles, Proceedings of the 5th International Symposium on Magnetic Bearings, Kanazawa, Japan, August 28-30, [10] Fittro, Roger L., Knospe, C.R., and Stephens, L.S., Experimental Results of mu Synthesis Applied to Point Compliance Minimization, Proceedings of the 5th International Symposium on Magnetic Bearings, Kanazawa, Japan, August 28-30, Compliance (mm/n) Exp. Theory Frequency (rad/sec) Figure 7: Dynamic compliance with PD controller Compliance (mm/n) Exp. Theory Frequency (rad/sec) Figure 8: Dynamic compliance with µ controller