- Email: [email protected]
Implementation of a fast tool servo with repetitive control for diamond turning
Mechatronics 13 (2003) 243–257
Implementation of a fast tool servo with repetitive control for diamond turning Marc Crudele, Thomas R. Kurfess
*
School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA Received 7 February 2000; accepted 25 April 2001
Abstract The integration of a fast tool servo (FTS) with piezoelectric actuation on a diamond turning machine (DTM) is presented. The intent of this research is to monitor improvements in surface quality in the areas of surface roughness and waviness observed in the facing of an aluminum work piece. Performance is compared between the utilization of a polynomial controller with position feedback of the tool, a repetitive controller that uses position feedback of the tool in conjunction with position feedback of the spindle face via and eddy current probe, and legacy performance of the DTM with no active control methodology. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction Diamond turning machines (DTMs) are well known for their exceptional surface finish generation ability. This is due, in part, to their large mass that helps to make the machine tool rigid. Cutting capability of the diamond is the other major factor that produces what is generally considered to be the best, non-abrasive method to obtain mirror finishes. Diamond turning has an advantage over abrasive methods due to its ability to retain precise, desired part geometry. 1.1. DTMs Most DTMs consist of a pair of orthogonal axes that are used to position the tool with the work piece mounted to the spindle. As the work piece rotates, the *
Corresponding author. Tel.: +1-404-894-0301; fax: +1-404-894-9342. E-mail address: [email protected] (T.R. Kurfess).
0957-4158/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 7 - 4 1 5 8 ( 0 1 ) 0 0 0 3 6 - 8
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orthogonal axes position the tool and feed it such that the piece can be faced or turned. High precision components are a must in diamond turning. The spindle must be well balanced to minimize vibration. The slide ways must be straight with reference to the spindle and must also be smooth in travel. Diamond tools are very brittle and any inconsistent force from vibrations, ‘jerks’, or interrupted cuts in the process can result in a damaged tool. Feed rates and depths are generally kept very low, making diamond turning mostly a finishing process. Ductile, non-ferrous metals and plastics are the usual work pieces. The tools used are mostly monocrystalline diamonds. Diamond tools have a long life – if properly used – due to their hardness, and they have a very low coefficient of friction as well as resistance to developing a build up edge. These factors, when coupled with a rigid machine tool lead to high quality surface finishes [1]. 1.2. Fast tool servos In the past, machines have taken a passive role in the quality of surface finish; they were made massive in order to reduce the sensitivity to vibrations. Speeds and feeds could be altered to avoid resonant frequencies and to try to achieve desired surface finishes, but beyond that, the machine tool had a limitation on its performance due to system vibration, thermal changes, and environmental disturbances. In order to achieve higher precision, one solution is to improve the quality of the hardware, which Rasmussen et al. [2] noted, affects the cost exponentially. With increasing computer capabilities, however, it has become possible to actively improve performance of existing machines by implementing devices such as fast tool servos (FTS). FTSs actuate the tool to counteract the detrimental effects present in the machining process. Because machine tools are traditionally massive, the inertia involved makes high-speed actuation of the slide ways impractical. If, however, a secondary actuation is implemented in the direction of interest, the slide way motors could be used in the traditional role of traversal/feed and the secondary actuation could limit itself to high speed control of a less massive cutting component over a smaller range (Fig. 1). By using a piezoelectric stack and the increased processing power of computers, this procedure has gained popularity in high precision applications. 1.3. Repetitive control Repetitive control – a form of learning control – is based on the simple premise that a controller can be designed to identify certain periodic patterns in a process and to improve the system’s performance by learning to expect these patterns. Implementations of this method are present, most notably, in robotics. A robotic system often performs repetitious tasks such as assembly. Another implementation is in tracking of a periodic set point. The controller will examine the commanded set point and issue a signal to the servo system. The controller will also monitor the
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Fig. 1. Concept of FTS.
system’s response and alter the signal until the output of the system tracks the command signal. As with humans, performance is expected to improve as the repetition of the task increases. 1.4. Objective of research Improving performance is a driving force in research. Development of FTSs has made strides in adding value through improving surface roughness of existing machine tools, but has mostly failed to address the problem of waviness as a function of the spindle speed and harmonics. The object of this research is to add a FTS to the existing DTM and to use polynomial control design with tool position feedback for active disturbance rejection. Once this is done, a learning controller with spindle face position feedback is implemented in order to address the issue of periodic spindle disturbances due to imbalances present in the system and to demonstrate the flexibility of the FTS. These imbalances resulted in work piece waviness that the learning controller is well suited to counteract. The remainder of this paper is ordered in the following manner. Section 2 will discuss the experimental setup and some of the equipment used. A theoretical discussion of the polynomial and repetitive controller will be presented in Section 3. Section 4 will present and analyze the results. The paper is concluded with a reevaluation of the FTS implementation and a discussion of what future applications could be explored.
2. FTS setup and equipment The setup for this research can be divided into three parts: the DTM, the FTS, and the supporting equipment.
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2.1. DTM The FTS was implemented on a DTM with a Moore Special Tool Company #3 Friction base with an Ex-Cell-O 40.64 cm (16 in.) air spindle. This utilized ElectroCraft DC brushless motors to control motion on the X and Z slide ways and a Scientific R43HB, brushless DC servo motor with a SC453 servo controller to control the spindle (Fig. 2). A PMAC card from Delta Tau in a Dell 120 MHz Pentium Computer provided control for the DTM. The card used has a Motorola DSP56001 CPU with a 40 MHz clock with a servo update frequency of 2.26 kHz. The PMAC option 28 card was also used for A/D conversion. Three tools are used in this research. A carbide insert with a nose radius of 0.794 mm (1/32 in.) is used for the initial cuts of the work piece. A PCD tool with a nose radius of 0.794 mm (1/32 in.) is then used to improve the surface finish. Finally, a radius nose, natural diamond turning tool with a radius of 0.762 mm (0.03 in.) and a 2.5° positive rake angle is used. 2.2. FTS The FTS was added in place of the pre-existing tool mount and is shown in Fig. 3. A Del-Tron RS2-6 crossed roller slide assembly with published straight line accuracy of 1 lm/cm of travel and repeatability of 1 lm with an aluminum base and carriage and hardened steel rods and rollers was used to obtain stability in translation while minimizing added actuated mass. Aluminum c-blocks are mounted to the slide and serve to hold the tool in place. A Physik Instrumente
Fig. 2. DTM.
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Fig. 3. FST.
P-244.47 piezotranslator with a nominal expansion of 60 lm, an open loop stiffness of 33 N=lm, an assembled pre-load of 350 N, a pushing force rating of 1000 N, an unloaded resonant frequency of 5.5 kHz, and strain gauge feedback was used as the actuation device. The PZT was secured to an external mount that was fastened to the original tool base through a slotted hole that clamps the PZT by using two tightening screws. The actuator was positioned against the linear slide under a spring pre-load. 2.3. Supporting equipment The DS1102 controller board from dSPACE Inc. was used to implement both the polynomial and repetitive controllers for the FTS. It was also used for collecting performance data. This card has a TI C31 DSP with a clock speed of 60 MHz, 2–16 bit ADCs, 2–12 bit ADCs, 4–12 bit DACs. dSPACE’s support software – Real Time Interface (RTI) – was used in conjunction with MATLAB to monitor and control the FTS. Also, MATLAB’s system identification toolbox was used to model the FTS for each of the tools described above. The data used for Matlab’s system identification came from Siglab – a system identification product – manufactured by DSP Technology. The hardware is composed of two inputs and two outputs with 20 kHz bandwidths, a 70 dB CMNR from DC to 500 Hz, 18 bit ADCs and 18 bit DACs. The equipment has a virtual sine wave generator that can be used in a variety of ways (including signal averaging chirp excitation) to gather system information. In order to monitor the spindle face position, a Kaman KD2300-2S eddy current probe is used. This is held in place by a tripod beside the DTM and is positioned
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radially from the cutting point of the work piece. The signal is scaled based on the position of the tool in reference to the center of the spindle face (provided by PMAC) in order to represent the amplitude of spindle face displacement as a function of radial distance from center. A Krohn-Hite 3384-12, tunable, active filter with eight-pole Butterworth filters is used to provide AC-coupling for the PZT strain gauge’s signal. This was necessary due to the drifting of the position signal and this research’s desire to track and counteract the higher frequency disturbances encountered by the tool as opposed maintaining an absolute position. The Krohn-Hite was also used to low pass the eddy current probe. A Zygo New View 200 scanning white-light interferometer is used to determine the roughness (RMS value) of each work piece by taking the average RMS reading of 20 measurements (five measurements at 90° intervals). The waviness (or angular parallelism) was measured using an TSK Rondcom 30B. 3. System modeling and controller design 3.1. System models The experimental model was determined by using Siglab’s virtual network analyzer module to derive the system’s frequency response. The command signal to the PZT amplifier has a range of 0–10 V; a setpoint signal (DC) of 5 V was sent to the PZT, which, due to the AC-coupling of the strain gauge, became the measured zero position. Data were collected on the input voltage (command plus 5 V setpoint) to the PZT and the position response as indicated by the voltage of the strain gauge. This provides both the frequency response and time data for the system by using a chirp signal for the input. The time data were imported to MATLAB’s system identification toolbox and were used to estimate the system model using arx, assuming a model less than seventh order. The procedure was repeated for each tool and it was discovered that any improvements obtained by models higher than third order were negligible. The modeled frequency responses in comparison to the experimental frequency responses are shown in Fig. 4 with the input being the command signal around 5 V and the output being the voltage of the strain gauge. Controllers were designed for each tool separately – due to their varying weights – based on these models. 3.2. Polynomial control design The polynomial control design methodology was very similar for each of the tools. Fig. 5 shows the basic design of the polynomial controller where B and A represent the numerator and denominator of the plant, respectively, in the form shown in Eq. (1)
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Fig. 4. Modeled vs. experimental bode plots.
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Fig. 5. Block diagram of polynomial design.
AðzÞ ¼ zn þ a1 zn1 þ þ an1 z þ an ;
ð1Þ
BðzÞ ¼ b0 zn þ b1 zn1 þ þ bn1 z þ bn :
The feed-back and feed-forward controller blocks are designed based on the desired closed-loop ðDðzÞÞ and observer ðKðzÞÞ characteristic equations. The other controller parameters (a and b) can then be calculated. This result is outlined in the following derivation: Gcl ¼
ðKB=aAÞ KB ¼ ; 1 þ ðbKB=KaAÞ aA þ bB
ð2Þ
DðzÞ ¼ aA þ bB
ð3Þ
with D, a, and b in the form: DðzÞ ¼ d0 z2n1 þ d1 z2n2 þ þ d2n2 z þ d2n1 ; aðzÞ ¼ a0 zn1 þ a1 zn2 þ þ an2 z þ an1 ; bðzÞ ¼ b0 z
n1
þ b1 z
n2
ð4Þ
þ þ bn2 z þ bn1 :
Eq. (3) is known as the Diaphantine equation and can be solved for a and b by using a 2n 2n Sylvester matrix E shown in Eq. (5) 2
an 6 an1 6 6 . 6 .. 6 6 6 6 a1 6 E¼6 1 6 6 0 6 6 . 6 . 6 . 6 4 0 0
0 an
... ...
0 0
an1 .. . a1 1 .. . 0 0
...
0 .. .
... ... ... ... ... ...
an1 an2 .. . a1 1
bn bn1 .. . b1 b0 0 .. . 0 0
0 bn
... ...
bn1 .. . b1 b0 .. . 0 0
... ... ... ... ... ... ...
3 0 0 7 7 7 0 7 7 .. 7 7 . 7 7 : bn1 7 7 7 bn2 7 .. 7 7 . 7 7 b1 5 b0
ð5Þ
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If D and M are then defined as: 3 2 an1 6 an2 7 3 2 7 6 d2n1 6 .. 7 7 6 . 6 d2n2 7 7 6 7 6 7 6 a 6 .. 7 0 7 6 D ¼ 6 . 7; M ¼ 6 7; b 7 6 6 n1 7 4 d1 5 6 bn2 7 7 6 d2 6 .. 7 4 . 5 b0
251
ð6Þ
then the coefficients of a and b can be solved for by modifying Eq. (5) to be M ¼ E1 D:
ð7Þ
With all of the control blocks defined, the controller was implemented on the DS1102 controller board and tested. 3.3. Repetitive control design A fundamental part of the repetitive controller is the use of a zero-phase error tracking controller (ZPETC). The discussion that follows is based on a paper published by Tomizuka [3] and information presented by Sadegh [4]. Ideal tracking with zero phase error is obtained by using a method of pole/zero cancellation and phase cancellation. The first step was to assume a stabilized, closed loop system – such as the one designed for the FTS with the polynomial control method. In order to achieve perfect tracking, it is desired to add a controller ðQÞ between the desired signal ðyd Þ and the closed loop plant as shown in Fig. 6. This controller should modify the desired signal into a form ðrÞ such that when it is sent through the closed loop servo transfer function, the output ðyÞ will equal the commanded signal. The obvious way to do this is to make Q the inverse of the closed loop servo. This would result in total pole/zero cancellation and therefore ideal tracking. In fact, this model is known as a perfect tracking controller (PTC). This concept is attractive but problematic if the servo system is non-minimum phase (zeros lie outside or on unit circle). If such a controller was implemented, the output of the controller could ‘explode’ due to its unstable nature. In this case, B can be separated into B ¼ Bþ ðzÞB ðzÞ such that Bþ ðzÞ represents minimum phase zeros
Fig. 6. Control structure concept for PTC.
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and B ðzÞ represents non-minimum phase zeros and the controller can be modified to that shown in Fig. 7 [3]. Further analysis of this controller structure shows the phase error at steady state is zero and unity gain exists at low frequencies. The bandwidth of this performance is dependent upon the system and can be analyzed using the frequency response [4]. 3.3.1. Implementing delay With an understanding of the ZPETC, the final steps of designing the repetitive controller can be examined. Fig. 8 shows the complete design of the system with Q as the ZPETC (previously discussed), Kl as the learning gain, and N being determined by using Eq. (8), where L is the period of the desired fundamental frequency and T is the sampling time L ð8Þ N¼ : T This model is based on the assumption that yd is a periodic command signal. QðzÞ was designed based on the closed loop servo H ðzÞ. This method assumes that H ðzÞ is asymptotically stable and has no zeros on the unit circle, kl > 0, QðzÞ has no poles on the unit circle, and zN QðzÞ must be proper (causal). To understand this control scheme, it is important to examine several components. Eq. (9) shows the closed loop transfer function for the system Y kl QH : ¼ N Yd z 1 þ kl QH
ð9Þ
The equation above must be stable. Eq. (10) shows a derivation of the error. It is important to remember that yd ðk þ N Þ ¼ yd ðkÞ because the desired input is periodic.
Fig. 7. Modified ZPETC controller.
Fig. 8. Model for repetitive control design.
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This means the error approaches zero as long as the closed loop system is asymptotically stable E Y zN 1 ; ¼1 ¼ N Yd Y z 1 þ kl QH d 1 eðkÞ ¼ N ½yd ðk þ N Þ yd ðkÞ : z 1 þ kl QH
ð10Þ
4. Results Utilizing the previously outlined theories resulted in the following results. 4.1. Surface roughness Fig. 9 shows a comparison of the RMS surface finish before the FTS, with the FTS using the polynomial controller, and with the FTS using the repetitive controller. There is no data in the above figure for the carbide tool using the repetitive controller due to instabilities that arose in that combination – possibly due to the controller’s inability to compensate for the increased force involved with cutting with a carbide tool. However, the surface roughness was reduced to 78.1% its original value by making the transition to the FTS (244–190.5 nm) and utilizing the polynomial controller. It can be seen that the PCD tool’s surface roughness was reduced to 65.7% its original value using the polynomial controller (115–75.5 nm) and 53.9% its original value using the repetitive controller (115–62 nm). The improvement in the performance of the repetitive controller over the polynomial controller is unexpected since the repetitive controller builds the learning component on the polynomial controller in order to address waviness. This is something that could warrant future investigation.
Fig. 9. RMS comparison of results.
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The radius nose diamond tool shows that the polynomial controller reduced the surface roughness to 81.8% of its pre-FTS value (27.5–22.5 nm). The repetitive controller reduced the surface roughness from the original configuration by 83.6% (27.5–23 nm). 4.2. Waviness The real advantage of the repetitive controller is shown in its ability to track the commanded periodic signal derived from the eddy current probe’s monitoring of the spindle face. This tracking ability is shown in Fig. 10. In the case of the PCD tool, the polynomial controller showed a RMS error of 0.0091 V and the repetitive
Fig. 10. Tracking comparisons of polynomial and repetitive controllers.
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controller had a RMS error of 0.0029 V (31.9% of polynomial). For the radius nose diamond, the RMS error of the polynomial controller is 0.0063 V and the repetitive controller’s RMS error is 0.0024 V (38.1% of polynomial). The benefit of the repetitive controller’s tracking ability is shown in the analysis of work piece waviness. This is shown in Fig. 11. The units of UPR are logical since the fundamental frequency can be thought of as the spindle frequency. The higher UPRs represent the harmonics of this fundamental frequency and decay exponentially in all cases, as would be expected. From the discussion of the surface finishes, both the polynomial and repetitive controller proved to be beneficial. However, the above figures show that the polynomial controller was detrimental to the waviness, increasing the amplitude at the fundamental frequency by 29.8% (8.76–11:37 lm) in the PCD case and 8.8% (10.92– 11:88 lm) in the radius nose diamond case. The repetitive controller showed similar improvement to surface finish and here is shown to reduce the waviness of the work piece at 1 UPR. In the PCD case, the waviness is reduced to 61.6% its original value (8.76–5:40 lm). The amplitude of the
Fig. 11. Waviness components.
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waviness using the radius nose diamond is reduced to 70.1% its original value (10.92– 7:73 lm). It can also be noted that these improvements do not carry over to the higher frequencies. This can be explained, in part, by the phase shift induced in the eddy current probe’s signal due to the need to low pass a 100 Hz signal present from 12 mounting holes on the spindle face in the area of monitoring.
5. Conclusion Improved precision in machining has been the source of much research. Traditional limitations of the cost involved in upgrading machine tool components and the difficulty of using real time control due to system inertia and computational intensity have been preventative. It is these limitations that encourage innovative designs such as FTSs. Reducing the range of actuation and mass coupled with taking advantage of improved processing power lead to the success of FTSs in this research. Using this technology and polynomial control design, improvement of surface finish of the carbide tool (244–190.5 nm), PCD tool (115–75.5 nm), and radius nose diamond tool (27.5–2.5 nm), were achieved. Implementation of the repetitive control scheme lead to a reduction of waviness in the pcd and radius nose diamond work pieces such that they were 60% and 72% their previous values, respectively. The reason for the improved surface roughness of the PCD tool with repetitive control (to 62 nm) is not apparent. The repetitive controller was expected to only improve waviness. This might warrant future investigation. The current setup not only improved the facing performance of the DTM, but also has added flexibility to the machine tool. Increased control over tool position allows both the possibility of improved traditional performance and the opportunity to explore more elegant, high performance applications such as generation of nonaspherically symmetric surfaces and microstructures. It is this versatility that is of special interest to producers of high precision and complex lenses and mirrors. The open architecture of the dSPACE card with its ability to interact with MATLAB and Simulink makes the implementation of any future controllers almost seamless. Thus, the success of this project has been in implementing the FTS, demonstrating the effectiveness of the system by improving surface roughness using polynomial control, and showing the flexibility of the FTS by beginning to address the issue of waviness using repetitive control.
Acknowledgements The author would like to take this opportunity to thank the engineers at The Torrington Company Manufacturing Development Center with special thanks to John Davidson. This work was partially funded by the National Science Foundation under grant nos. DDM-9350159, DMI-9622692 and DMI-9812967. The government
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has certain rights in this material. Any opinions, findings and conclusions or recommendations are those of the authors and do not necessarily reflect the views of the National Science Foundation or The Torrington Company.
References [1] Ulmer B. Fabrication and calibration of an open architecture diamond turning machine, Thesis for MS degree in Mechanical Engineering at Georgia Institute of Technology, GA, 1997. [2] Rasmussen JD et al. Dynamic variable depth of cut machining using piezoelectric actuators. Int J Mach Tools Manufact 1994;16(1). [3] Tomizuka M. Zero phase error tracking algorithm for digital control. J Dyn Systems, Measurement, Control March 1987;109. [4] Sadegh N. Class Notes for ME6438. Atlanta, GA: Georgia Institute of Technology, Mechanical Engineering Department; 1998.
Implementation of a fast tool servo with repetitive control for diamond turning Marc Crudele, Thomas R. Kurfess
*
School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA Received 7 February 2000; accepted 25 April 2001
Abstract The integration of a fast tool servo (FTS) with piezoelectric actuation on a diamond turning machine (DTM) is presented. The intent of this research is to monitor improvements in surface quality in the areas of surface roughness and waviness observed in the facing of an aluminum work piece. Performance is compared between the utilization of a polynomial controller with position feedback of the tool, a repetitive controller that uses position feedback of the tool in conjunction with position feedback of the spindle face via and eddy current probe, and legacy performance of the DTM with no active control methodology. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction Diamond turning machines (DTMs) are well known for their exceptional surface finish generation ability. This is due, in part, to their large mass that helps to make the machine tool rigid. Cutting capability of the diamond is the other major factor that produces what is generally considered to be the best, non-abrasive method to obtain mirror finishes. Diamond turning has an advantage over abrasive methods due to its ability to retain precise, desired part geometry. 1.1. DTMs Most DTMs consist of a pair of orthogonal axes that are used to position the tool with the work piece mounted to the spindle. As the work piece rotates, the *
Corresponding author. Tel.: +1-404-894-0301; fax: +1-404-894-9342. E-mail address: [email protected] (T.R. Kurfess).
0957-4158/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 7 - 4 1 5 8 ( 0 1 ) 0 0 0 3 6 - 8
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orthogonal axes position the tool and feed it such that the piece can be faced or turned. High precision components are a must in diamond turning. The spindle must be well balanced to minimize vibration. The slide ways must be straight with reference to the spindle and must also be smooth in travel. Diamond tools are very brittle and any inconsistent force from vibrations, ‘jerks’, or interrupted cuts in the process can result in a damaged tool. Feed rates and depths are generally kept very low, making diamond turning mostly a finishing process. Ductile, non-ferrous metals and plastics are the usual work pieces. The tools used are mostly monocrystalline diamonds. Diamond tools have a long life – if properly used – due to their hardness, and they have a very low coefficient of friction as well as resistance to developing a build up edge. These factors, when coupled with a rigid machine tool lead to high quality surface finishes [1]. 1.2. Fast tool servos In the past, machines have taken a passive role in the quality of surface finish; they were made massive in order to reduce the sensitivity to vibrations. Speeds and feeds could be altered to avoid resonant frequencies and to try to achieve desired surface finishes, but beyond that, the machine tool had a limitation on its performance due to system vibration, thermal changes, and environmental disturbances. In order to achieve higher precision, one solution is to improve the quality of the hardware, which Rasmussen et al. [2] noted, affects the cost exponentially. With increasing computer capabilities, however, it has become possible to actively improve performance of existing machines by implementing devices such as fast tool servos (FTS). FTSs actuate the tool to counteract the detrimental effects present in the machining process. Because machine tools are traditionally massive, the inertia involved makes high-speed actuation of the slide ways impractical. If, however, a secondary actuation is implemented in the direction of interest, the slide way motors could be used in the traditional role of traversal/feed and the secondary actuation could limit itself to high speed control of a less massive cutting component over a smaller range (Fig. 1). By using a piezoelectric stack and the increased processing power of computers, this procedure has gained popularity in high precision applications. 1.3. Repetitive control Repetitive control – a form of learning control – is based on the simple premise that a controller can be designed to identify certain periodic patterns in a process and to improve the system’s performance by learning to expect these patterns. Implementations of this method are present, most notably, in robotics. A robotic system often performs repetitious tasks such as assembly. Another implementation is in tracking of a periodic set point. The controller will examine the commanded set point and issue a signal to the servo system. The controller will also monitor the
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245
Fig. 1. Concept of FTS.
system’s response and alter the signal until the output of the system tracks the command signal. As with humans, performance is expected to improve as the repetition of the task increases. 1.4. Objective of research Improving performance is a driving force in research. Development of FTSs has made strides in adding value through improving surface roughness of existing machine tools, but has mostly failed to address the problem of waviness as a function of the spindle speed and harmonics. The object of this research is to add a FTS to the existing DTM and to use polynomial control design with tool position feedback for active disturbance rejection. Once this is done, a learning controller with spindle face position feedback is implemented in order to address the issue of periodic spindle disturbances due to imbalances present in the system and to demonstrate the flexibility of the FTS. These imbalances resulted in work piece waviness that the learning controller is well suited to counteract. The remainder of this paper is ordered in the following manner. Section 2 will discuss the experimental setup and some of the equipment used. A theoretical discussion of the polynomial and repetitive controller will be presented in Section 3. Section 4 will present and analyze the results. The paper is concluded with a reevaluation of the FTS implementation and a discussion of what future applications could be explored.
2. FTS setup and equipment The setup for this research can be divided into three parts: the DTM, the FTS, and the supporting equipment.
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2.1. DTM The FTS was implemented on a DTM with a Moore Special Tool Company #3 Friction base with an Ex-Cell-O 40.64 cm (16 in.) air spindle. This utilized ElectroCraft DC brushless motors to control motion on the X and Z slide ways and a Scientific R43HB, brushless DC servo motor with a SC453 servo controller to control the spindle (Fig. 2). A PMAC card from Delta Tau in a Dell 120 MHz Pentium Computer provided control for the DTM. The card used has a Motorola DSP56001 CPU with a 40 MHz clock with a servo update frequency of 2.26 kHz. The PMAC option 28 card was also used for A/D conversion. Three tools are used in this research. A carbide insert with a nose radius of 0.794 mm (1/32 in.) is used for the initial cuts of the work piece. A PCD tool with a nose radius of 0.794 mm (1/32 in.) is then used to improve the surface finish. Finally, a radius nose, natural diamond turning tool with a radius of 0.762 mm (0.03 in.) and a 2.5° positive rake angle is used. 2.2. FTS The FTS was added in place of the pre-existing tool mount and is shown in Fig. 3. A Del-Tron RS2-6 crossed roller slide assembly with published straight line accuracy of 1 lm/cm of travel and repeatability of 1 lm with an aluminum base and carriage and hardened steel rods and rollers was used to obtain stability in translation while minimizing added actuated mass. Aluminum c-blocks are mounted to the slide and serve to hold the tool in place. A Physik Instrumente
Fig. 2. DTM.
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Fig. 3. FST.
P-244.47 piezotranslator with a nominal expansion of 60 lm, an open loop stiffness of 33 N=lm, an assembled pre-load of 350 N, a pushing force rating of 1000 N, an unloaded resonant frequency of 5.5 kHz, and strain gauge feedback was used as the actuation device. The PZT was secured to an external mount that was fastened to the original tool base through a slotted hole that clamps the PZT by using two tightening screws. The actuator was positioned against the linear slide under a spring pre-load. 2.3. Supporting equipment The DS1102 controller board from dSPACE Inc. was used to implement both the polynomial and repetitive controllers for the FTS. It was also used for collecting performance data. This card has a TI C31 DSP with a clock speed of 60 MHz, 2–16 bit ADCs, 2–12 bit ADCs, 4–12 bit DACs. dSPACE’s support software – Real Time Interface (RTI) – was used in conjunction with MATLAB to monitor and control the FTS. Also, MATLAB’s system identification toolbox was used to model the FTS for each of the tools described above. The data used for Matlab’s system identification came from Siglab – a system identification product – manufactured by DSP Technology. The hardware is composed of two inputs and two outputs with 20 kHz bandwidths, a 70 dB CMNR from DC to 500 Hz, 18 bit ADCs and 18 bit DACs. The equipment has a virtual sine wave generator that can be used in a variety of ways (including signal averaging chirp excitation) to gather system information. In order to monitor the spindle face position, a Kaman KD2300-2S eddy current probe is used. This is held in place by a tripod beside the DTM and is positioned
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radially from the cutting point of the work piece. The signal is scaled based on the position of the tool in reference to the center of the spindle face (provided by PMAC) in order to represent the amplitude of spindle face displacement as a function of radial distance from center. A Krohn-Hite 3384-12, tunable, active filter with eight-pole Butterworth filters is used to provide AC-coupling for the PZT strain gauge’s signal. This was necessary due to the drifting of the position signal and this research’s desire to track and counteract the higher frequency disturbances encountered by the tool as opposed maintaining an absolute position. The Krohn-Hite was also used to low pass the eddy current probe. A Zygo New View 200 scanning white-light interferometer is used to determine the roughness (RMS value) of each work piece by taking the average RMS reading of 20 measurements (five measurements at 90° intervals). The waviness (or angular parallelism) was measured using an TSK Rondcom 30B. 3. System modeling and controller design 3.1. System models The experimental model was determined by using Siglab’s virtual network analyzer module to derive the system’s frequency response. The command signal to the PZT amplifier has a range of 0–10 V; a setpoint signal (DC) of 5 V was sent to the PZT, which, due to the AC-coupling of the strain gauge, became the measured zero position. Data were collected on the input voltage (command plus 5 V setpoint) to the PZT and the position response as indicated by the voltage of the strain gauge. This provides both the frequency response and time data for the system by using a chirp signal for the input. The time data were imported to MATLAB’s system identification toolbox and were used to estimate the system model using arx, assuming a model less than seventh order. The procedure was repeated for each tool and it was discovered that any improvements obtained by models higher than third order were negligible. The modeled frequency responses in comparison to the experimental frequency responses are shown in Fig. 4 with the input being the command signal around 5 V and the output being the voltage of the strain gauge. Controllers were designed for each tool separately – due to their varying weights – based on these models. 3.2. Polynomial control design The polynomial control design methodology was very similar for each of the tools. Fig. 5 shows the basic design of the polynomial controller where B and A represent the numerator and denominator of the plant, respectively, in the form shown in Eq. (1)
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Fig. 4. Modeled vs. experimental bode plots.
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Fig. 5. Block diagram of polynomial design.
AðzÞ ¼ zn þ a1 zn1 þ þ an1 z þ an ;
ð1Þ
BðzÞ ¼ b0 zn þ b1 zn1 þ þ bn1 z þ bn :
The feed-back and feed-forward controller blocks are designed based on the desired closed-loop ðDðzÞÞ and observer ðKðzÞÞ characteristic equations. The other controller parameters (a and b) can then be calculated. This result is outlined in the following derivation: Gcl ¼
ðKB=aAÞ KB ¼ ; 1 þ ðbKB=KaAÞ aA þ bB
ð2Þ
DðzÞ ¼ aA þ bB
ð3Þ
with D, a, and b in the form: DðzÞ ¼ d0 z2n1 þ d1 z2n2 þ þ d2n2 z þ d2n1 ; aðzÞ ¼ a0 zn1 þ a1 zn2 þ þ an2 z þ an1 ; bðzÞ ¼ b0 z
n1
þ b1 z
n2
ð4Þ
þ þ bn2 z þ bn1 :
Eq. (3) is known as the Diaphantine equation and can be solved for a and b by using a 2n 2n Sylvester matrix E shown in Eq. (5) 2
an 6 an1 6 6 . 6 .. 6 6 6 6 a1 6 E¼6 1 6 6 0 6 6 . 6 . 6 . 6 4 0 0
0 an
... ...
0 0
an1 .. . a1 1 .. . 0 0
...
0 .. .
... ... ... ... ... ...
an1 an2 .. . a1 1
bn bn1 .. . b1 b0 0 .. . 0 0
0 bn
... ...
bn1 .. . b1 b0 .. . 0 0
... ... ... ... ... ... ...
3 0 0 7 7 7 0 7 7 .. 7 7 . 7 7 : bn1 7 7 7 bn2 7 .. 7 7 . 7 7 b1 5 b0
ð5Þ
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If D and M are then defined as: 3 2 an1 6 an2 7 3 2 7 6 d2n1 6 .. 7 7 6 . 6 d2n2 7 7 6 7 6 7 6 a 6 .. 7 0 7 6 D ¼ 6 . 7; M ¼ 6 7; b 7 6 6 n1 7 4 d1 5 6 bn2 7 7 6 d2 6 .. 7 4 . 5 b0
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ð6Þ
then the coefficients of a and b can be solved for by modifying Eq. (5) to be M ¼ E1 D:
ð7Þ
With all of the control blocks defined, the controller was implemented on the DS1102 controller board and tested. 3.3. Repetitive control design A fundamental part of the repetitive controller is the use of a zero-phase error tracking controller (ZPETC). The discussion that follows is based on a paper published by Tomizuka [3] and information presented by Sadegh [4]. Ideal tracking with zero phase error is obtained by using a method of pole/zero cancellation and phase cancellation. The first step was to assume a stabilized, closed loop system – such as the one designed for the FTS with the polynomial control method. In order to achieve perfect tracking, it is desired to add a controller ðQÞ between the desired signal ðyd Þ and the closed loop plant as shown in Fig. 6. This controller should modify the desired signal into a form ðrÞ such that when it is sent through the closed loop servo transfer function, the output ðyÞ will equal the commanded signal. The obvious way to do this is to make Q the inverse of the closed loop servo. This would result in total pole/zero cancellation and therefore ideal tracking. In fact, this model is known as a perfect tracking controller (PTC). This concept is attractive but problematic if the servo system is non-minimum phase (zeros lie outside or on unit circle). If such a controller was implemented, the output of the controller could ‘explode’ due to its unstable nature. In this case, B can be separated into B ¼ Bþ ðzÞB ðzÞ such that Bþ ðzÞ represents minimum phase zeros
Fig. 6. Control structure concept for PTC.
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and B ðzÞ represents non-minimum phase zeros and the controller can be modified to that shown in Fig. 7 [3]. Further analysis of this controller structure shows the phase error at steady state is zero and unity gain exists at low frequencies. The bandwidth of this performance is dependent upon the system and can be analyzed using the frequency response [4]. 3.3.1. Implementing delay With an understanding of the ZPETC, the final steps of designing the repetitive controller can be examined. Fig. 8 shows the complete design of the system with Q as the ZPETC (previously discussed), Kl as the learning gain, and N being determined by using Eq. (8), where L is the period of the desired fundamental frequency and T is the sampling time L ð8Þ N¼ : T This model is based on the assumption that yd is a periodic command signal. QðzÞ was designed based on the closed loop servo H ðzÞ. This method assumes that H ðzÞ is asymptotically stable and has no zeros on the unit circle, kl > 0, QðzÞ has no poles on the unit circle, and zN QðzÞ must be proper (causal). To understand this control scheme, it is important to examine several components. Eq. (9) shows the closed loop transfer function for the system Y kl QH : ¼ N Yd z 1 þ kl QH
ð9Þ
The equation above must be stable. Eq. (10) shows a derivation of the error. It is important to remember that yd ðk þ N Þ ¼ yd ðkÞ because the desired input is periodic.
Fig. 7. Modified ZPETC controller.
Fig. 8. Model for repetitive control design.
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This means the error approaches zero as long as the closed loop system is asymptotically stable E Y zN 1 ; ¼1 ¼ N Yd Y z 1 þ kl QH d 1 eðkÞ ¼ N ½yd ðk þ N Þ yd ðkÞ : z 1 þ kl QH
ð10Þ
4. Results Utilizing the previously outlined theories resulted in the following results. 4.1. Surface roughness Fig. 9 shows a comparison of the RMS surface finish before the FTS, with the FTS using the polynomial controller, and with the FTS using the repetitive controller. There is no data in the above figure for the carbide tool using the repetitive controller due to instabilities that arose in that combination – possibly due to the controller’s inability to compensate for the increased force involved with cutting with a carbide tool. However, the surface roughness was reduced to 78.1% its original value by making the transition to the FTS (244–190.5 nm) and utilizing the polynomial controller. It can be seen that the PCD tool’s surface roughness was reduced to 65.7% its original value using the polynomial controller (115–75.5 nm) and 53.9% its original value using the repetitive controller (115–62 nm). The improvement in the performance of the repetitive controller over the polynomial controller is unexpected since the repetitive controller builds the learning component on the polynomial controller in order to address waviness. This is something that could warrant future investigation.
Fig. 9. RMS comparison of results.
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The radius nose diamond tool shows that the polynomial controller reduced the surface roughness to 81.8% of its pre-FTS value (27.5–22.5 nm). The repetitive controller reduced the surface roughness from the original configuration by 83.6% (27.5–23 nm). 4.2. Waviness The real advantage of the repetitive controller is shown in its ability to track the commanded periodic signal derived from the eddy current probe’s monitoring of the spindle face. This tracking ability is shown in Fig. 10. In the case of the PCD tool, the polynomial controller showed a RMS error of 0.0091 V and the repetitive
Fig. 10. Tracking comparisons of polynomial and repetitive controllers.
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controller had a RMS error of 0.0029 V (31.9% of polynomial). For the radius nose diamond, the RMS error of the polynomial controller is 0.0063 V and the repetitive controller’s RMS error is 0.0024 V (38.1% of polynomial). The benefit of the repetitive controller’s tracking ability is shown in the analysis of work piece waviness. This is shown in Fig. 11. The units of UPR are logical since the fundamental frequency can be thought of as the spindle frequency. The higher UPRs represent the harmonics of this fundamental frequency and decay exponentially in all cases, as would be expected. From the discussion of the surface finishes, both the polynomial and repetitive controller proved to be beneficial. However, the above figures show that the polynomial controller was detrimental to the waviness, increasing the amplitude at the fundamental frequency by 29.8% (8.76–11:37 lm) in the PCD case and 8.8% (10.92– 11:88 lm) in the radius nose diamond case. The repetitive controller showed similar improvement to surface finish and here is shown to reduce the waviness of the work piece at 1 UPR. In the PCD case, the waviness is reduced to 61.6% its original value (8.76–5:40 lm). The amplitude of the
Fig. 11. Waviness components.
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waviness using the radius nose diamond is reduced to 70.1% its original value (10.92– 7:73 lm). It can also be noted that these improvements do not carry over to the higher frequencies. This can be explained, in part, by the phase shift induced in the eddy current probe’s signal due to the need to low pass a 100 Hz signal present from 12 mounting holes on the spindle face in the area of monitoring.
5. Conclusion Improved precision in machining has been the source of much research. Traditional limitations of the cost involved in upgrading machine tool components and the difficulty of using real time control due to system inertia and computational intensity have been preventative. It is these limitations that encourage innovative designs such as FTSs. Reducing the range of actuation and mass coupled with taking advantage of improved processing power lead to the success of FTSs in this research. Using this technology and polynomial control design, improvement of surface finish of the carbide tool (244–190.5 nm), PCD tool (115–75.5 nm), and radius nose diamond tool (27.5–2.5 nm), were achieved. Implementation of the repetitive control scheme lead to a reduction of waviness in the pcd and radius nose diamond work pieces such that they were 60% and 72% their previous values, respectively. The reason for the improved surface roughness of the PCD tool with repetitive control (to 62 nm) is not apparent. The repetitive controller was expected to only improve waviness. This might warrant future investigation. The current setup not only improved the facing performance of the DTM, but also has added flexibility to the machine tool. Increased control over tool position allows both the possibility of improved traditional performance and the opportunity to explore more elegant, high performance applications such as generation of nonaspherically symmetric surfaces and microstructures. It is this versatility that is of special interest to producers of high precision and complex lenses and mirrors. The open architecture of the dSPACE card with its ability to interact with MATLAB and Simulink makes the implementation of any future controllers almost seamless. Thus, the success of this project has been in implementing the FTS, demonstrating the effectiveness of the system by improving surface roughness using polynomial control, and showing the flexibility of the FTS by beginning to address the issue of waviness using repetitive control.
Acknowledgements The author would like to take this opportunity to thank the engineers at The Torrington Company Manufacturing Development Center with special thanks to John Davidson. This work was partially funded by the National Science Foundation under grant nos. DDM-9350159, DMI-9622692 and DMI-9812967. The government
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has certain rights in this material. Any opinions, findings and conclusions or recommendations are those of the authors and do not necessarily reflect the views of the National Science Foundation or The Torrington Company.
References [1] Ulmer B. Fabrication and calibration of an open architecture diamond turning machine, Thesis for MS degree in Mechanical Engineering at Georgia Institute of Technology, GA, 1997. [2] Rasmussen JD et al. Dynamic variable depth of cut machining using piezoelectric actuators. Int J Mach Tools Manufact 1994;16(1). [3] Tomizuka M. Zero phase error tracking algorithm for digital control. J Dyn Systems, Measurement, Control March 1987;109. [4] Sadegh N. Class Notes for ME6438. Atlanta, GA: Georgia Institute of Technology, Mechanical Engineering Department; 1998.