Application of a fast tool servo for diamond turning of nonrotationally symmetric surfaces

Application of a fast tool servo for diamond turning of nonrotationally symmetric surfaces T h o m a s A. D o w , * M i c h e l e H. M i l l e r * and P e t e r J. F a l t e r t • Precision Engineering Center, North Carolina State University, USA, t Westinghouse Defense and Electronic Systems, Orlando, FL, USA

The fabrication of nonrotationa/ly symmetric surfaces by diamond turning requires too/actuation at a bandwidth significantly higher than the rotational frequency of the surfaces. This requirement cannot be met by standard sfide drives due to their large mass and consequent low natural frequency. This article describes the development of a laboratory-scale diamond-turning machine with piezoelectric-driven fast tool servo. The capability of this apparatus will be demonstrated for high-speed features such as sine wave, square wave, and ramp-shaped surfaces. Also described is the implementation of this fast too/servo on a commercial diamond-turning machine. Several nonrotationa//y symmetric surfaces have been machined, and their images are included. K e y w o r d s : diamond turning; PZT actuators; fast too/servo; nonrotationa#y symmetric surfaces; reference surface; spindle gro wth correction, real- time control

Introduction

High-precision fabrication has become increasingly important to manufacturing and the world economy. Consumer items such as video cassette recorders and compact disk players contain components machined to tolerances that were once found only on research and military equipment. While high-precision items become more common in everyday applications, the defense and scientific communities are constantly challenging US precision manufacturing capabilities. In response, new technologies must continually advance to machine tomorrow's more difficult materials and geometries both accurately and efficiently. Single point diamond turning is an evolving machining technology that generates the most accurate and repeatable geometries and the finest surface finishes of any general machining method. While often used to produce exotic or highly demanding one-of-a-kind elements, diamond turning is being applied to higher volume items including infrared optics, computer memory disks, contact lenses, lens molds, print drums, scanner

Address reprint requests to Thomas A. Dow, Precision Engineering Center, North Carolina State University, Raleigh, NC 27695, USA.

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mirrors, transducers, and optical sensors. 1 The capability of modern diamond-turning machines to produce aspheric elements has had a large impact on the optics industry. 2 Both complete and off-axis aspheric surfaces can be machined, including geometries that are difficult and prohibitively expensive to produce by the more traditional grinding and polishing techniques used in optical shops. However, current diamond-turning techniques cannot be applied to the generation of every type of aspheric surface. Like other turning techniques, they have been limited to surfaces of revolution, i.e., surfaces that do not vary as a function of rotation. A truly general asphere might possess rotational asymmetries that would render it nonproducible via current diamond-turning technology. The commercial production of nonrotationally symmetric surfaces could have a similar effect on optical system design and performance as was experienced when diamond-turned aspheres first became available. Possible applications include wavefront corrector plates for large optical systems, 3 scanning mirrors, optical encoders, extreme off-axis reflectors, and specialty lenses and molds with intentional asymmetries such as coma and astigmatism. This technology might also produce extreme off-axis forms in an on-center fashion 4 6 as

© 1991 Butterworth-Heinemann

243

Dow et a l . Fast tool servo for diamond turning well as mechanical elements such as cams and pistons. Fabrication of nonrotationally symmetric surface components requires extremely high resolution and bandwidth but only a small range of nonsymmetry. It is for this application that a piezoelectric-driven fast tool servo (FTS) has been developed for a diamond-turning machine. This article describes the development of a servo to fabricate nonrotationally symmetric optical surfaces. Two diamond-turning machines are discussed: the first is a laboratory-scale machine based on a pair of parallel axis air-bearing spindles, and the second is a larger commercial diamond-turning machine with a T-base design. The dynamics of the machine, the controller used to ameliorate these dynamics, and the results of experiments machining nonrotationally symmetric surfaces are described.

Parallel axis ultraprecision lathe

FTS. A DC motor-driven lead screw is used to push the cutting arm across the surface of the workpiece in a shallow arc. A coarse adjustment of the depth of cut can be made using a differential screw, which holds the FTS to the cutting arm, and fine adjustment is made with the FTS Fast t o o l servo. The details of the FTS are illustrated in Figure 2. The heart of the servo ~s a hollow piezoelectric actuator (25 mm OD and 18 mm long) with a maximum range of 20/lm. This element is preloaded by a set of round, plate-like flexures that are integral with the cutting tool mount. The moving mass, including the tool and its mount, is approximately 140 g. The natural frequency of the FTS, which is a function of the moving mass and the stiffness of the piezoelectric stack, is approximately 10 kHz. The open-loop frequency response function of the FTS is illustrated in Figure 3. The response is flat with an amplitude

A laboratory-scale diamond-turning machine capable of fabricating nonrotationally symmetric surfaces in metallic materials has been described. 7' 8 The basic design of the diamond-turning machine including the FTS, workpiece-based metrology, and high-speed digital servo control were discussed. What appears here is a summary of the relevant features of that apparatus and some experimental results.

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M e c h a n i c a l s t r u c t u r e . The parallel axis ultraprecision lathe (PAUL) apparatus is shown in Figure 1. It consists of a pair of 100-mm air bearings, with vertical axes mounted on a rigid, horizontal steel base. One bearing supports the workpiece and is driven by an integral DC motor. The second bearing holds the cutting arm with the

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1 PAUL with FTS for fabrication of nonrotationally symmetric optical surfaces 244

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Figure 3 Frequency response function of FTS from 0 to 5 kHz OCTOBER 1991 VOL 13 NO 4

Dow et al.. Fast tool servo for diamond turning ratio of only 1 db at 2 kHz with minimal phase lag. This means that the FTS can follow a sine wave at 2 kHz with only 12% error in amplitude. M e t r o l o g y s y s t e m . The metrology system consists of a pair of high-bandwidth capacitance (ADE Microprox 2 1 0 2 / 2 0 3 6 ) gauges and a moving reference surface. The capacitance gauges were independently calibrated using a laser interferometer. The first gauge is integral with the FTS and is used to measure the motion of the tool relative to the cutting arm. The second gauge is mounted on the cutting arm and measures the relative position of this arm and a reference surface attached to the workpiece. The diamond-turned reference surface is shown in Figure 1. The controller computes the instantaneous depth of cut from these two gauge outputs. A m p l i f i e r . A high-power amplifier was designed and built to drive the FTS. This unit was designed to produce voltage and current sufficient to move the piezoelectric element to its full extension (_+ 10/~m) at quasistatic conditions and about one fourth of its full extension ( _+ 2.5/~m) at 1 kHz. The resulting amplifier design consists of a push section of 14 power transistors in series that supply current to the piezoelectric element and a similar pull section that drains current from the actuator. Even though the amplifier is capable of continuous operation, the piezoelectric element cannot be run at a constant high frequency because of internal heat generated by the losses in the crystal. For the conditions reported in this article, the frequency was low enough that no problems were encountered; but if necessary, a cooling system could be developed to overcome this problem.

Control system S y s t e m m o d e l . The proper selection of a control system for the PAUL requires an understanding of the geometry of the apparatus, its dynamic characteristics, and displacement feedback from the sensors in the metrology system. The major mechanical system to be controlled is the FTS itself. However, because of its stiff design and small moving mass, the dynamic characteristics are easy to control over the bandwidth commonly needed for optical fabrication. The ancillary components include the servo drive electronics, metrology equipment, and filters. The geometry of the apparatus determines the relationship between the motion of the capacitance gauge in the FTS and the gauge on the reference surface. C o m p u t e r h a r d w a r e . The high-speed control calculations for the FTS were performed on a digital signal processor (TI TMS320C25). This chip has a 100 ns cycle time and an instruction set optimized for the computations required to implement filters, controllers, and other real-time applications. It is part of a commercial data acquisition board (Ariel PRECISION ENGINEERING

DSP-16) that includes two channels of A / D , two channels of D/A, and PC Bus interface electronics.

Experimental results The results of the fabrication experiments using the PAUL/FTS are discussed for two modes of operation: (1) quasistatic correction for spindle growth errors, and ( 2 ) dynamic operation for the fabrication of nonrotationally symmetric optical surfaces. S p i n d l e g r o w t h c o r r e c t i o n . Spindle growth is a form of thermal distortion found in hydrostatic, aerostatic, and mechanical bearing spindles. It is a low-frequency error, with a time constant on the order of minutes. Since the amount of heat generated varies with the square of spindle speed, spindle growth can become significant in high-speed applications. For diamond-turning, spindle growth can cause significant errors in parts machined before the spindle has warmed up. Experiments were made using the PAUL/FTS apparatus with and without closed-loop feedback for spindle compensation. With feedback from the reference surface, the errors due to spindle thermal effects were reduced by 85% to <0.1 /~m. F o l l o w i n g e r r o r . Following error is the error in position of the tool compared to the reference signal. An example of closed-loop following error for a 2.5/~m peak-to-valley, 40-Hz sine wave is shown in Figure 4. The large sine wave is the reference waveform and the smaller signal (at higher magnification) is the difference between this reference and the motion of the piezoelectric actuator. The following error is a maximum when the reference signal has its maximum velocity, which is a consequence of the proportional/integral (PI) control scheme. The maximum following error for this condition is _+30 nm or about 2% of the input amplitude. This value, which is an order of

Figure4 Following error ( 6 0 n m ) of tool compared to reference sine wave (2.5/~m peak-to-valley) 245

Dow et al.. Fast tool servo for diamond turning magnitude larger than the open-loop following error, is a consequence of the cycle time of the controller and the control algorithm selected. Disturbance rejection. Disturbance rejection is a measure of the control system's ability to compensate for noise, unmodeled dynamics, and other disturbances. The unmodeled hysteresis of the piezoelectric actuator, for example, has been successfully overcome through the use of the feedback capacitance gauge. Structural vibrations are another important error source in precision machining. Such disturbances may be generated by machine elements or the cutting process or may be transmitted through the floor or the air. To study the disturbance rejection capability of the PAUL/FTS, an electromagnetic shaker was attached to the base of the machine stand and was used to produce vertical disturbances at 40, 140, and 90 Hz--below, above, and at the primary mode of vibration of the cutting arm (90 Hz). Cutting experiments were performed on a brass workpiece with the controller on and off, and the peak-to-valley roughness of the surfaces produced were measured. These experiments are summarized in Table 1, which shows the amplitude of the surface roughness as measured on the Talysurf profilometer in the feed direction. The average reduction in surface roughness was 78%. These results illustrate the enormous benefit from this control system for eliminating the effect of structural disturbances on the machined surface. Surface features. Several surfaces were machined with nonrotationally symmetric features in the form of sine waves and triangle waves. To fabricate these surfaces a reference shape was input to the controller, and feedback from the metrology system was used to correct errors in the shape. Typical of these surfaces is a copper specimen with ten 2.5-i~m saw-tooth shaped waves per revolution. The interferogram of this surface is shown in Figure 5. It was machined at 200 rpm and the corresponding sinusoidal frequency was 33 Hz. The general shape of the surface--i.e., the straightness of the fringes and the number from peak to valley--could be

F i g u r e 5 Interferogram of ramp surface with 2.5 l~m peak-to-valley amplitude

determined from the interferogram, but detailed information on the distortion of the machined surface could not be obtained To overcome this difficulty, a different technique of inspecting the surface in the form of circular profiles at a given radius was used, and the results are discussed in the following section. Figure 6 shows a profilomete~ trace of severai waves of a saw-tooth shaped surface with 40 waves per revolution measured with a modified roundness profilometer. Also shown in this figure is the ideal reference saw--tooth used to comnqand the toot. The cutting speed was 100 rpm so that the fundamentai frequency for the saw-tooth pattern is 67 Hz. From the trace, the repeatability of the peaks is evident as is the reduction in amplitude of the features from the 2.5 ym peak-to-valley magnitude of the reference surface to 2 ym. The rounding of the waveform and the resulting reduction in the height occurred as a result of the control algorithm gains used during this test. The PI control algorithm

Table 1 Disturbance rejection of structural vibrations Rejection of structural vibration Amplitude (nm) Frequency

Control on

Control off

% Reduction

40 90 140

63 127 102

380 610 356

83 79 71

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Figure 6 Ideal reference profile and fabricated ramp surface with peak-to-valley amplitude of 2 ~nq and 40 waves per revolution OCTOBER 1991 VOL 13 NO 4

Dow et al. : Fast tool servo for diamond turning modifies the high-frequency regions of the command to avoid excitation at the natural frequency of the FTS. As a result, the closed-loop bandwidth of the FTS is reduced, and the sharp features of the reference shape are rounded onto the measured surface profile. This profile was similar to the motion of the tool as measured by the FTS capacitance gauge when the tool was not in contact with the workpiece. This indicates that the cutting forces have much less effect than expected on the shape of the resulting surface.

T-base d i a m o n d - t u r n i n g machine Based on the results of the PAUL experiments, the FTS was added to a commercial two-axis T-base diamond-turning machine. The Z coordinate then has two components: an axisymmetric component assigned to the Z slide and a high-frequency nonaxisymmetric component produced by the FTS (Z'). For the results described here, the FTS controller is independent of the diamond-turning machine axes controller. So that the X position can be an input for the FTS position control, these two controllers are being integrated using a parallel processing architecture, Heterogenous Hierarchical Architecture for Real-Time (H2ART), 9 which has been developed at the Precision Engineering Center over the past several years.

Definition of nonrotationa/ly symmetric surface geometry A critical element of FTS implementation is developing a method for representing nonrotationally symmetric surfaces. For rotationally symmetric parts the axial dimension Z is a function only of the radial dimension X. For example, the reference generation equation Z = X 2 would produce a paraboloid. For nonrotationally symmetric parts Z has angular dependence. An example of a surface with both angular and radial dependence is Z = X s i n nO. In this case, Z is calculated based on angular position of the spindle (~) provided by a spindle encoder as well as X position from a laser interferometer. The number of points required to accurately describe a surface of the form Z = f(X, O) is approximately N e times greater than that required for Z = f ( X ) (where Ne is the number of encoder pulses per revolution). In addition, the rate at which the points are used increases correspondingly. To address this computer controller challenge, two reference generation approaches have been considered. One approach is to calculate all Z' positions off-line and to store them in memory for sequential execution during the cutting process. The disadvantage to this approach is that it requires a vast amount of storage space for each surface. The second approach is to calculate Z' positions on-the-fly during machining. The disadvantage of this method is that it limits the complexity of the surface description equation. As the following PRECISION ENGINEERING

machining examples show, both approaches have been used.

O-Dependent surfaces For a generic ~)-dependent surface, the FTS moves in the same pattern for each spindle revolution. If a new Z' value is output only on each encoder pulse interrupt, then the number of Z' values needed to machine the entire surface is equal to the number of encoder pulses per revolution. A table of all Z' values is therefore small enough to reside in the data memory of the DSP. An example of a surface that can be machined using a short table is Z'(l~m) = 0.3 sin 100

(1)

If machined with a 2,000 count encoder, this surface requires only a 200 point table that cycles 10 times each revolution. To machine this surface, a table of output values is prepared off-line and inserted into the DSP control program. The DSP algorithm consists of sensing an encoder pulse interrupt, reading a table value, outputting a voltage to the FTS, and indexing the table pointer. A minimum cycle time of 5.3 Fs is required to execute this algorithm on a TI TMS320C25 processor, which translates into a maximum cutting speed of 5,660 rpm (assuming a 2,000 count spindle encoder). Figure 7 shows an interferogram and part of a

B F i g u r e 7 Zygo Mark IV images of 0.6/~m peak-to-valley circumferential sine waves 247

D o w et al.. Fast tool servo for diamond turning

topographical plot for a surface described by Equation (1), which was machined on a 63.5-mm diameter copper workpiece at 500 rpm. This method of constructing tables also has usefulness when evaluating the trigonometric functions included in descriptions of surfaces with R and 8 dependence. Although entries will not be in the form of output values, complete decimal tables for each trigonometric function in the reference generation equation can be developed that require no interpolation between values. R-~) D e p e n d e n t surfaces Assuming a constant feed rate and spindle speed, any surface with both radial and angular dependence can be converted into a surface with only angular dependence by using the linear relationship between R (or X) and ~). The tool traverses a spiral path with respect to the workpiece, and for a cut that starts at X Rwpand finishes at X = 0, X is related to ~ by =

f X = Rwp - - 0 T s

(2)

where Rwp = f = s= 0T =

radius of workpiece feed rate spindle speed number of revolutions since start of cut

The advantage of removing the radial dependence is that the FTS DSP does not have to read the X laser interferometer every controller cycle. The disadvantage is that tool centering errors, X slide following error, and spindle speed error introduce errors into the surface. The FTS and a TI TMS320C30 floating point DSP have been used to machine surfaces of the form Z = aXsin n& X position was approximated using Equation (2), and sin n~? was read from a table according to the method described previously. The C30 processor required a minimum cycle time of 34.5 #s (maximum cutting speed of 870 rpm) to complete the algorithm. Figure 8 is a phase map and topographical plot showing sine waves (at 10 cycle/rev frequency) that shrink from a peak-to-valley of 1.25/~m at the outside edge to 0 #m at the center. F i x e d - d e p t h features Another type of R-~ dependent surface that can be machined by making use of Equation (2) is a surface with fixed-depth features. Since only two cutting depths are used for this type of cut, preprocessing can reduce the number of points to a manageable number for storage. The tool moves in or out where the cutting spiral intersects line segments that define a figure. The () increments between intersection points can be found off-line and downloaded as an array to the DSP. During

248

B

Figure 8 Zygo Mark IV images of Z ~ aR sin 10~

cutting the DSP counts encoder pulses, compares the count to the next number in the array, and increments or decrements the voltage to the FTS when appropriate. The minimum cycle time required to execute this algorithm on a TMS320C25 is 4.8 #s, which translates into a maximum cutting speed of 6,250 rpm. Figure 9 shows an interferogram and topographical plot of a 0.25/~m deep angstrom symbol /~, which was machined at 500 rpm on a 63.5-mm diameter copper workpiece. Although effective surface measurement techniques have not yet been developed for nonrotationally symmetric surfaces, machining surfaces with fixed-depth features proved useful in showing the sensitivity of all nonrotationally symmetric surfaces to specific machine error sources. Some of the initial cuts produced skewed figures caused by tool centering errors. The effects of Y and X centering errors were determined analytically and simulated. An error in the Y direction warps a square figure as shown in Figure 10,4. An error in the X direction causes figure edges to bow as shown in Figure lOB. Both effects were observed in machined parts and corrected by adjusting the tool center. Two other causes of the bowing effect shown in B are X slide following error and inaccurate spindle speed. X slide following error has the same effect as the tool zero being set short of OCTOBER 1991 VOL t 3 NO 4

D o w et al.. Fast tool servo for diamond turning

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center; however, due to the magnitude of the following error (on the order of 1 /~m), this is a small effect. The effect of spindle speed may be more significant. The preprocessing for each figure assumes a constant ratio of X slide feed rate to spindle speed. Because the spindle speed is uncontrolled, this ratio may vary from its intended value. A spindle speed slightly faster than prescribed would have the same effect as the tool becoming progressively short of center. Reading the position with the laser interferometer and using this in the control system would eliminate this source of error.

Summary The primary objective of this research has been to investigate the production of nonrotationally symmetric surfaces via diamond-turning. This PRECISION ENGINEERING

objective requires the development of machining, metrology, and control hardware as well as control software. Experiments have yielded important information about the producibility of such surfaces for optical and other applications. The tool servo has performed beyond expectations. The high stiffness and low moving mass combine to give a usable bandwidth of over 2 kHz. A capacitance gauge in the servo body provides feedback to correct both the hysteresis and drift in the piezoelectric actuator. A high-current amplifier was designed and built to drive the capacitance load of the piezoelectric crystal at the high frequency desired for this application. The amplifier design worked well and was capable of driving the servo well beyond the design goal of 1 kHz. The metrology frame was intended to measure and compensate for any differences in motion of the tool in the FTS body and in the workpiece. The experiments indicated that the dynamic thrust forces due to cutting had little influence on the shape of the surface and that the reference surface for that purpose was not needed. However, it was very 249

D o w et al. " Fast t o o l servo f o r d i a m o n d t u r n i n g

useful for compensating for the thermal spindle growth and other large-amplitude, low-frequency disturbances. Unfortunately, using a reference surface is only practical when machining flats. One of the most challenging problems in fabricating nonrotationally symmetric elements is verification of the final geometry. Since such surfaces are not widely used, commercially available systems designed to measure truly amorphic surfaces do not exist. Commercial laser interferometers cannot measure high spatial frequency variations. Multiple wavelength and white light techniques may be useful, but a large aperture version is not currently available. Computergenerated holograms may prove effective for null waveform generation for interferometric testing of nonrotationally symmetric surfaces. Mechanical measurements are also possible; e.g., a long-range profilometer with rotary table option might be used to perform high-accuracy inspection of many nonrotationally symmetric surfaces. Unfortunately, damage to the surface from stylus contact is a distinct possibility with this technique. A high-speed integrated controller is needed to fabricate nonrotationally symmetric surfaces. The results presented here use an add-on DSP-based controller with encoder and capacitance gauge feedback for the FTS position. However, information about the X and Z slide position is also needed for ultimate shape control. Currently, such an integrated controller is being built and programmed.

250

Acknowledgments This work was supported in part by the Office of Naval Research University Research Initiative Program in Precision Engineering and the corporate affiliates of the Precision Engineering Center~

References 1

Langenbeck, P "Precision diamond machining of metal optics A review." Optoe/ektronik. 1984 2 Saito, T. T "Precision machining application and technology An overview and perspective." Proc. SP/E 1983, 433 3 Schenz, R. F., Patterson, S. R. and Saito, T. T. "Direct machining of a non-axisymmetric phase corrector." Proc SP/E

1988, 966, 43 59 4 Douglass,S. S. "A machining systemfor turning nonaxisymmetric surfaces." Ph.D. diss., University of Tennessee, 1983 5 Gerchman, M. C. " A n exact description of far off-axis comc surfaces for non-rotationally symmetric surface generation ' Proceedings of the AS P E Annual Meeting, 1989, pp. 104 107 6 Thompson, D. C. "Theoretical tool movement required to diamond turn an off axis parabotoid on axis " Proc. SPIE 1976, 93, 2 3 - 2 9 7 Falter, P J and Dow, T. A. "Adiarnond-turning apparatus fo~" fabrication of non-rotationally symmetric surfaces," Proceedings of the International Congress for Ultraprecision Technology, Aachen, Germany, May 1988, Springer Verlag, pp. 187 -. 20t 8 Falter, P. J, "Diamond turning of nonrotationally symmetnc surfaces." Ph.D. diss., North Carolina State University, 1990 9 Fornaro, R. J., Garrard, K. P. and Taylor, L. W. "Architecture and algorithms for computer control of high precision machine tools.'" Proceedings of the ASPE Annual Meeting, 1990, pp 27 30

OCTOBER 1991 VOL 13 NO 4.