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Analysis of thermal errors in a high-speed micro-milling spindle
ARTICLE IN PRESS International Journal of Machine Tools & Manufacture 50 (2010) 386–393
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International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Analysis of thermal errors in a high-speed micro-milling spindle E. Creighton, A. Honegger, A. Tulsian, D. Mukhopadhyay n Microlution Inc., 4038 N Nashville Ave, Chicago, IL 60634, USA
a r t i c l e in f o
a b s t r a c t
Article history: Received 16 March 2009 Received in revised form 1 November 2009 Accepted 2 November 2009 Available online 11 November 2009
Thermally induced errors account for the majority of fabrication accuracy loss in an uncompensated machine tool. This issue is particularly relevant in the micro-machining arena due to the comparable size of thermal errors and the characteristic dimensions of the parts under fabrication. A spindle of a micro-milling machine tool is one of the main sources of thermal errors. Other sources of thermal errors include drive elements like linear motors and bearings, the machining process itself and external thermal influences such as variation in ambient temperature. The basic strategy for alleviating the magnitude of these thermal errors can be achieved by thermal desensitization, control and compensation within the machine tool. This paper describes a spindle growth compensation scheme that aims towards reducing its thermally-induced machining errors. The implementation of this scheme is simple in nature and it can be easily and quickly executed in an industrial environment with minimal investment of manpower and component modifications. Initially a finite element analysis (FEA) is conducted on the spindle assembly. This FEA correlates the temperature rise, due to heating from the spindle bearings and the motor, to the resulting structural deformation. Additionally, the structural deformation of the spindle along with temperature change at its various critical points is experimentally obtained by a system of thermocouples and capacitance gages. The experimental values of the temperature changes and the structural deformation of the spindle qualitatively agree well with the results obtained by FEA. Consequently, a thermal displacement model of the high-speed micro-milling spindle is formulated from the previously obtained experimental results that effectively predict the spindle displacement under varying spindle speeds. The implementation of this model in the machine tool under investigation is expected to reduce its thermally induced spindle displacement by 80%, from 6 microns to less than 1 micron in a randomly generated test with varying spindle speeds. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Micro-milling Thermal compensation High-speed spindle Micro-machining Spindle growth model
1. Introduction Many fields ranging from bio-medical devices [1] to defense applications [2] are increasing using parts and devices with micron-scale characteristic dimensions. Miniature scale devices provide significant advantages over their macro-scale counterparts in terms of cost, size and performance [3]. Over the last few years these devices have tremendously grown in the complexity of their functions and level of performance which has directly impacted their design features, and has led to a need in increase of their manufacturing accuracy. To address this need for increased accuracy of manufacturing micro-scale parts and devices, machine tool builders have greatly improved the performance of their machining centers. However, dimensional errors arising due
n
Corresponding author. E-mail address: [email protected] (D. Mukhopadhyay).
0890-6955/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2009.11.002
to temperature variations that distort various machine tool components, defined as thermal errors, account for a majority of loss in fabrication accuracy of thermally uncompensated machine tools [4]. Moreover, the feature sizes in micro-scale parts are of the same order as these thermal errors. Hence, the effects of these thermal errors are more pronounced in micro-scale machining. A spindle of a micro-milling machine tool is one of the main sources of thermal errors [5–7] along with drive elements such as linear motors and bearings, the machining process itself and external thermal influences like the variation in ambient temperature [7]. There are three primary methods for counteracting these thermal errors which can be categorized as: elimination or avoidance, resistance and control, and compensation tactics. Elimination or avoidance tactics tend to completely eliminate any change in dimensions due to thermal fluctuations. They are best (or rather can only be) implemented during the design phase of the machine tool. Some examples of these tactics are
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symmetric design and serpentine design of machine tool structures. Resistance and control tactics tend to reduce and/or oppose any change in dimensions due to thermal fluctuations. They are best implemented during the build phase of the machine tool. Some examples of these tactics are use of thermally insensitive materials and highly efficient components that generate very low heat in machine tool components. Compensation tactics tend to compensate for any change in dimensions due to thermal fluctuations. They can be implemented during any phase of the machine tool design and development. Some examples of these tactics include passive modeling of thermal deformation machine tool models and active thermal compensation using thermocouples and thermal-structural mapping functions in machine tool control software. The elimination and control tactics are generally preferred over the compensation tactics. However, under many circumstances implementation of a compensation scheme is required. These circumstances can include: improving the accuracy of a machine tool after it has been built and reducing the residual thermal errors after the implementation of elimination and control tactics. Extensive amount of research has been conducted towards developing various compensation tactics geared specifically towards combating the thermal error due to spindle displacement or deformation. Researchers have used multi-variant regression analysis [8–10], thermal error mode [11,12], artificial neural network [13], dynamic simulation [5] and others techniques to develop precise compensation schemes to counteract spindle deformation due to thermal effects. Some of these schemes are developed offline and then implemented on a machine tool as an internal spindle deformation model [9], where as others are implemented a real time basis [14,15]. In general, the majority of the work done previously has concentrated on using medium to large spindles to implement the developed compensation tactics. A micro-milling spindle differs significantly from a larger spindle. The differences are not only in relatively size and power consumption but also in maximum speeds and distribution of its power between the cutting process and the power needed to overcome the friction of a spindle’s bearings. A clear proportional relationship between the thermal growth of a spindle and its speed has been established by other researchers [8–13]. Since the maximum speed of a micro-milling spindle (upwards of 50,000 RPM) is generally higher than its larger counterpart (up to 20,000 RPM), this growth effect is expected to be relatively more significant for a high-speed micro-milling spindle. Additionally, in a larger spindle used in macro-machining most of the spindle power is used for cutting. However, in micro-machining only a small percentage of spindle’s power is used for cutting [15]; the rest is used to overcome the friction in its bearings. Hence, a significant portion of spindle thermal growth occurs during the heat generated in overcoming this friction. This portion of the spindle growth can be effectively counteracted by capturing the spindle deformation while running the spindle in free-air and developing a compensation scheme based on those growth values. Finally, the smaller spindles used for micro-milling are generally air cooled, since they do not have enough real-estate for installation of effective liquid-cooling channels. Since liquid cooling has a higher heat capacity compared to a similarly sized air-cooling channels, these micro-milling spindles may grow significantly even in the presence of cooling, thus necessitating a presence of a growth compensation scheme. Moreover, many of the small machine tools used for micromilling are used intermittently with significant number of starts and stops throughout the day. A implementation of a spindle displacement compensation scheme that not only predicts the steady-state growth at a specific speed but also the rate of growth
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(or the transient state of spindle growth) can lead to increased productivity by eliminating the ‘‘warm-up’’ time associated with many larger machines, in which the machine is warmed up for a specified amount of time and spindle speed to let the thermal growth stabilize before use. Hence, there is a clearly defined need for a high-speed micromilling spindle displacement compensation scheme that is relatively easy to implement within a short period of time, and is quite reliable in reducing the machining errors, that arise due to the thermal deformation of the micro-milling spindle. There have been many machine tools using medium to large spindles that use some sort of compensation scheme to counteract spindle growth. However, there is still a lack of use of spindle growth compensation schemes in many small machine tools that are geared towards micro-milling. This can be attributed to two main factors, the time and effort it takes to implement available compensation tactics on a machine tool in an industrial environment and the robustness of the implemented model [16]. This paper describes such a high-speed micro-milling spindle displacement compensation scheme that aims to reduce the spindle-associated thermal errors. The implementation of this scheme is simple in nature and it can be easily and quickly executed in an industrial environment with minimal investment of manpower and component modifications. This paper is organized as follows. Initially, a finite element analysis (FEA) is conducted on this spindle to determine the
Fig. 1. Schematic of spindle mounting; (a) isometric view and (b) side view showing the offset between the spindle centerline and the mounting points of its housing to the rest of the model (Note: the near-side linear motor and its coil are hidden in this view).
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temperature distribution of the spindle, motor and its housing. This temperature field is fed back into the FEA model to ascertain the resultant structural deformation at the tool-tip, which directly affects the accuracy of the micro-scale part that is fabricated by this spindle. The analysis results were compared against the experimentally measured temperatures of the setup and the deformation at the tool-tip at a specific speed. Additional experimental runs were conducted at various spindle speeds with setup temperatures and tool-tip deformations measured during each run. These experimental results were then extended to develop a micro-milling spindle displacement model and compensation scheme. Finally, this compensation scheme was applied to a random operating sequence of the spindle to demonstrate its effectiveness in reducing thermal errors due to the spindle deformation.
2. Finite element analysis A finite element analysis of the spindle growth under the heat generated by the motor and the bearings was conducted to get an approximate temperature distribution in the spindle, motor and its housing. The schematic of the setup (or the model) in which this analyses (and the later experimental tests) were conducted is shown in Fig. 1. In this setup, a NSK NR40-5100 high-speed micro-milling spindle with an EM3060 motor was used to develop the thermal compensation scheme (shown in Fig. 2). This spindle has two sets of ceramic ball bearings, one near the spindle nose and the other
near the back face of the spindle. Moreover the bearing near the spindle nose is 6.35 mm in width and the one near the back face of the spindle is 12.7 mm in width. The motor is attached to the spindle behind the rear set of bearings. Additionally, the motor itself has two sets of bearings, both 12.7 mm in width, one placed at the front end of the motor and the other at its back end. This spindle is held between its bearings by the clamp of the spindle housing. The spindle housing is then attached to the rest of the setup through its mounting points. Additionally, the motor is attached to the back of the spindle but is not supported on any other location. In order to approximately simulate the geometric thermal changes for the spindle, a brief static-thermal FEA of its structure was conducted. The spindle body was assumed to be made of solid steel. The base of the analysis setup is made of black granite and the rest of the structural components are made of steel. The FEA was conducted in ANSYS v11 (workbench module). The model was meshed into 9973 elements, both hexagonal and tetrahedral in shape (using the mixed meshing configuration). This level of meshing leads to a good tradeoff between the simulation accuracy and solution time. For gauging the temperature distribution in the model, heat-flux boundary conditions were applied to the major heat generating sources; the bearings (2.5e 3 W/mm2) and the motor of the spindle (5.5e 4 W/mm2). This bearing and motor heat load was calculated from the power drawn by the spindle while running freely at 50,000 RPM (approximately 17.5 W) and by assuming 80% efficiency for the spindle motor. Additionally, this spindle uses air-cooling channels through its body, to prevent the spindle from overheating when
Fig. 2. A NSK NR40-5100 high-speed micro-milling spindle with an EM3060 motor.
Fig. 3. (a) Temperature distribution of the analysis setup at 50,000 RPM; and (b) temperature distribution along the spindle (Note: spindle housing and clamp are hidden in this view).
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operating under high speeds. A forced air convection coefficient (of 1.5e 4 W/mm2) was applied along the spindle body [17] to simulate this cooling effect. Lastly, effect of heat dissipation from the entire setup was captured by applying a typical near still-air surface-to-air convection coefficient of 2.5e 5 W/mm2 [17] to all the setup surfaces exposed to air. Fig. 3 shows the result of the FEA including the temperature distribution in the spindle, motor and spindle housing under a spindle-operating speed of 50,000 RPM (while rotating freely in air). From Fig. 3(b) it can be deduced that spindle-motor junction has the highest temperature in the model, as this is the joining point of two main heat sources in this setup. As expected there is a slight temperature increase in the body of the motor. However, the spindle air cooling is effective in reducing the temperature of the spindle body as well as the front bearing. The rest of the model including the base, experience minimal temperature increases (approximately + 0.26 1C) due to the spindle and the motor heat conduction. This non-uniform elevated temperature distribution in the analysis model is consequently expected to result in tool-tip deformation that affects the accuracy of any small machine tool in which this spindle is used. To gage this tool-tip deformation, the previously obtained temperature distribution was fed back to a
Fig. 4. Deformation of the spindle, motor and housing due under thermal stresses.
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static-structural FEA setup on the same model. In this analysis, the reference temperature was set at 22 1C which was the same used in the static-thermal analysis. Additionally, the sides of the granite base (with normal along X axis) were structurally constrained in all directions. The initial ‘‘thermal-condition’’ of this analysis was specified to be the results of the previously conducted static-thermal analysis. Fig. 4 shows deformation of the model under this static-structural analysis with a thermal loading condition. From Fig. 4 it can be observed the tip of the spindle deforms by approximately 6.6 mm. This deformation is distributed both as an axial elongation, as well as a deformation along the positive Y direction. There is no net deformation along the X axis due to the symmetric design of the model about the Y–Z plane. The deformation along the positive Y direction is in the form of a tilt about the X axis. This is due to the temperature differences along the length of the spindle and the offset (in the Y-direction) between the center-line of the spindle and the mounting points of the housing [Fig. 1(b)].
3. Experimental setup The previously conducted FEA gave approximate values and locations of the expected high- and low-temperature zones, along with expected deformations at the tool tip. However, detailed experiments with the physical setup were conducted to develop a more accurate characterization of the thermal profile of the model and the resultant tool-tip deformations at various spindle speeds while rotating in free air. All these experimental tests used the same set-up as shown in Figs. 5 and 6. Several different sensors (including thermocouples and capacitive sensors) and data acquisition equipment were used in order to gather the data necessary for the spindle thermal characterization. The geometry of the spindle at the tool tip was measured axially through capacitive sensors (from Lion Precision). The temperature of the spindle and motor was measured through type T glass braided thermocouples (from Omega) at critical points. The actual spindle speed and the torque produced by the motor were fed back through the spindle motor controller in real time. Tables 1 and 2 show the specifications for the spindle, thermocouples, and capacitive sensors, respectively, that were used in the spindle characterization experiments. For these tests, thermocouple (TC) placements were determined by locating the thermal sources in the spindle and motor. The front of the spindle contains bearings that produce heat. One
Fig. 5. Schematic of the spindle thermal characterization experimental setup.
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thermocouple (TC #3) was placed at the top of the spindle to monitor this heat generation. The face of the spindle was in contact with the bearings and had two air vents used to cool the spindle. Two thermocouples were located in this area; one (TC #1) near an air vent, where the air would have the largest affect, and the other (TC #2) equidistance from the two air vents, where the air would have the smallest affect. Two thermo-
couples were placed at the connection between the spindle and the motor since heat is produced in driving the spindle; one thermocouple (TC #4) was placed at the back of the spindle and another (TC #5) at the front of the motor. The final thermocouple (TC #6) was placed at near the power supply at the back of the motor. See Fig. 7 for exact placement of the thermocouples. An additional thermocouple, (TC #7) was used to monitor the ambient temperature. The following tests were conducted to gage the tool-tip deformation under various spindle-operating speeds: 1. Run spindle at 10,000 rpm until temperature has stabilized (about 4 h). Stop spindle. 2. Force cool the spindle with compressed air, allowing it to return to previous starting temperature (about 1.5 h). 3. Repeat with spindle at 20,000, 30,000, 40,000, and 50,000 rpm. 4. Use data acquisition software to record data for the temperature and geometry of the spindle. Use CNC Data Acquisition program to record the spindle speed and torque during the test.
Fig. 6. Spindle thermal characterization experimental setup.
Table 1 Spindle specifications (NSK NR40-5100) for spindle thermal characterization. NSK NR40-5100 spindle specifications Maximum speed Air for Collet opening Spindle accuracy Weight Power consumption rating Maximum output power
50,000 RPM 0.55a–0.61 MPa o 2 mm [0.00008’’] 0.74 kg [1.63 bs] 1.8 A 350 W
Table 2 Thermocouple (Type T) and capacitive sensors specifications for spindle thermal characterization. Type T thermocouple specifications Tolerance A/D converter
1 1C or 0.75% High-resolution, 22-bit
Capacitive sensors specifications Bandwidth ( 3 dB) Peak-to-peak resolution RMS resolution Sensitivity
15,000 Hz 50 nm 10 nm 40 mV/mm
Fig. 8 shows the temperature reading from TC #1 for various operating speeds of the spindle for a single experimental run. Additionally, Fig. 9 shows the temperature readings from all the thermocouples at the maximum spindle speed of 50,000 RPM. The saw-tooth like variation in Fig. 8 and Fig. 9 was due to the cycling of the laboratory’s HVAC system. This can be clearly observed by noticing the cyclic variation (with a magnitude of 70.5 C) in the temperature values read by TC#7, which monitors the ambient temperature. The corresponding axial tool displacements as measured simultaneously by the capacitance sensors are shown in Fig. 10. From this graph it can be noticed that the time taken to reach the steady state is independent of the spindle speed and the magnitude of the steady-state values increases non-linearly with the spindle speed. The cyclic variations in the displacement values are due to the corresponding variations in the ambient temperature as mentioned earlier. The experimental spindle thermal characterization followed the finite element analysis, qualitatively both in terms of temperature distribution and the tool-tip deformation. The experimental data shows that under air-cutting conditions the highest temperature for the spindle was at the spindle motor junction (TC #4 and #5) due to the heat generated by the motor and lowest was at the front of the spindle (TC #1), mostly due to the convection cooling caused by the air flow through the spindle. This matched with the finite element analysis results [Fig. 3(b)] of the spindle body. Also, the data shows that at a spindle speed of 50,000 RPM, a typical micro-machining spindle speed, the axial tool displacement was approximately 5.3 mm. However, this displacement was simulated as 6.6 microns [Fig. 4]. The
Fig. 7. Placement of thermocouples for the spindle characterization tests.
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discrepancy between these two values can be attributed to underestimation of the convection cooling of the air flow through the spindle, assumption in the values of various fluxes used in the simulation and the lack of extensive knowledge about the internal structure of spindle. Through these tests, several key observations were made. As expected, the axial tool displacement follows the spindle temperature, which itself was affected by the ambient temperature. The ambient temperature for the tests fluctuated causing variations in both the spindle temperature and the axial tool displacement. The maximum axial tool displacement was close to 6 mm. The spindle temperature and the axial tool displacement data from the tests show an exponential rise to a constant steady state, a trend that matches the exponential spindle growth model in [18]. This steady state varies depending on the spindle speed but the exponential time constant does not. The exponential rise of the spindle to its steady state took about 3000 s.
27 10,000 rpm 20,000 rpm 30,000 rpm 40,000 rpm 50,000 rpm
26
Temperature (C)
25
24
23
22
21
0
0.5
1
1.5
2
391
2.5
Time (sec)
x 104
4. Model development
Fig. 8. Temperature readings for various spindle speeds.
30 thermocouple 1 thermocouple 2 thermocouple 3 thermocouple 4 thermocouple 5 thermocouple 6 thermocouple 7
29
Temperature (C)
28 27 26 25 24 23 22 21 0
0.5
1
1.5
2
2.5 x 104
Time (sec)
Fig. 9. Temperature readings for various thermocouples at 50,000 RPM spindle speed.
The results from the characterization tests of the spindle under different operating speeds was used to develop a spindle growth model that is exponential in nature, as shown in (1) and similar to ones reported in [14,18].
Dz ¼ z0 þ ðzss z0 Þð1 et=t Þ
ð1Þ
where, Dz is the change in spindle growth over a time t, z0 is the spindle deformation at time 0, zss is the steady state spindle deformation for a particular speed and t is the time constant for the spindle growth (which is different from the time constant when the spindle is contracting during its slow down to a lower speed). For this model, steady state and time constant values were found from the conducted experimental tests. This data is summarized in Table 3. The rise time and fall time data was averaged to calculate two time constants; one for warm-up and the other for cool-down of the spindle. The next step was to develop an equation to determine the steady-state axial tool displacements at different spindle speeds. The axial tool growth data from multiple experimental runs were graphed against the actual spindle speeds and fitted to a straight line (Fig. 11). The linear fit between the spindle-thermal displacement and the actual spindle rotation speed was a good fit (2), (R value of approximately 0.95) and allows for continuous determination of the steady state (zss in microns) at various spindle speeds
7 10,000 rpm 20,000 rpm 30,000 rpm 40,000 rpm 50,000 rpm
6 5
micron
4
5.34µm 3.70µm
3 2
2.03µm
1
1.15µm 0.97µm
0 -1
Table 3 Axial tool displacement exponential curve data. Commanded speed (rpm)
50,000 40,000 30,000 20,000 10,000
-2 0
0.5
1
1.5
2
time (sec) Fig. 10. Axial tool growth for various spindle speeds.
Warm-up
Cool-down
Rise time (sec)
Steady state (mm)
Fall time (sec)
Steady state (mm)
803.3 706.9 750 668 900
5.34 3.70 2.03 1.15 0.97
540 470 530 448.4 305.8
5.34 3.35 2.2 1.35 0.7
Warm-up time constant
Cool-down time constant
695.38 51.71
518.82 40.62
2.5 x 104
Average Std. Dev.
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in the direction away from the spindle body thus leaving the tooltip position unchanged. This is due to the clamping arrangement of the spindle and the location of the motor behind the spindle body in which the motor is free to expand without effecting the position of the tool tip.
6 y = 1.1083E-4x - 0.48942
Axial Displacement (µm)
5
R2 = 0.9464
4
5. Results
3 2 1 0 -1 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 x
Speed (rpm)
5
A random speed step test was designed to gage the effectiveness of the exponential spindle growth model thus developed. Fig. 13 shows the profile of the stepped speed test. Capacitance gages were used to measure the actual spindle deformation under the stepped speed profile test. The spindle growth model was then used to predict the spindle growth under the speed profile. Fig. 14 shows the effect of the growth model in compensating for the spindle displacement in the random speed test. In this figure, the blue line denotes the tool-tip deformation (axial) as collected on a real time basis and the green line denotes
104
Fig. 11. Steady-state axial tool growth versus actual spindle speed.
4 Spindle Speed (rpm)
6
5 Axial Displacement (µm)
x 104
5
4
3
3
2
1
2
0
1
-1 0
1
2
3
4
5
6
7
8 x 104
time (sec) 0 0
5
10
15
20
Fig. 13. Stepped spindle speed test for testing the developed model.
25
Power (%) Fig. 12. Steady-state axial tool growth versus drawn spindle power.
6
(vspindle in RPM) that were not considered in the experimental runs. ð2Þ
Additional tests were conducted to verify if the axial growth mapped more accurately to power drawn by the spindle. For these tests, a brush with metal bristles was attached to the spindle instead of the tool to increase the torque produced by the spindle to rotate the brush at the same speed as the tool. This setup also captures the additional torque needed during an actual machining process. Fig. 12 shows the graph of axial growth vs. spindle power. It can be observed that there is no clear relationship between the spindle axial growth and the power drawn by the spindle. Hence, the axial growth of the spindle is more closely aligned with spindle speed than power. This observation can be explained as follows. When the torque needed by the spindle increases at a specific speed, the motor draws more power and consequently heats up. This additional heat generates a thermal load on the motor which overhangs from behind the spindle body. However, this thermal load is gaged to only elongate the body of the motor
Axial Thermal Growth (µm)
zss ¼ 1:11e4 vspindle 0:49
Speed Step Test Speed Step Model
5 4 3
6µm
2 1
1µm
0 -1 -2 0
1
2
3
4 Time (sec)
5
6
7
8 x 104
Fig. 14. Prediction of tool-tip deformation by the developed model.
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the predicted deformation of the tool-tip by the previously developed model. From Fig. 14 it can be observed that without implementing the spindle growth compensation scheme, the spindle growth can reach up to 6 mm. However, this growth can be reduced to 1 mm by implementing the developed compensation scheme (i.e. moving the Z-axis on a real-time basis by the value predicted by the model in the opposite direction), resulting in a reduction of 80% in errors due to thermal sources. Another point that can be noted from Fig. 14 is the initial jump in the tool-tip position. This can be attributed to the structural relaxation that the spindle bearings undergo when the spindle is initially switched on. Furthermore, this behavior is absent when the speed of the spindle is increased from a non-zero initial value.
6. Conclusions This paper presents a high-speed micro-milling spindle displacement compensation scheme that is based on the exponential growth model. The initial finite element analysis results match qualitatively with the experimentally observed growth data to form a simple compensation scheme that can be implemented in an industrial environment with minimal investment of time, material and personnel. The compensation scheme is able to reduce the effect of thermal errors by up to 80% under a random stepped speed test. To ensure that the above developed spindle thermal compensation scheme is robust under different application environments; several methods of compensation were additionally considered. These included using only the developed model, using only tool length checks using a laser sensor and using the developed model in conjunction with intermittent tool length checks. The first type of implementation, using only the developed model is less favored as compared to the last one as the error between the actual and the modeled axial tool displacement can increase once steady state is reached. This error can accumulate over time, leading to errors similar in magnitude to errors without implementing the compensation scheme. This error accumulation is the result of ambient temperature fluctuations and drift over time. Though ambient conditions will not be accounted for the model, allowing for intermittent tool length checks along with the use of the developed scheme will help to reduce these errors.
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Acknowledgements We thank the ONR, Office of Naval Research (Contract no N00014-08-C-0088) for funding a part of this research endeavor.
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International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Analysis of thermal errors in a high-speed micro-milling spindle E. Creighton, A. Honegger, A. Tulsian, D. Mukhopadhyay n Microlution Inc., 4038 N Nashville Ave, Chicago, IL 60634, USA
a r t i c l e in f o
a b s t r a c t
Article history: Received 16 March 2009 Received in revised form 1 November 2009 Accepted 2 November 2009 Available online 11 November 2009
Thermally induced errors account for the majority of fabrication accuracy loss in an uncompensated machine tool. This issue is particularly relevant in the micro-machining arena due to the comparable size of thermal errors and the characteristic dimensions of the parts under fabrication. A spindle of a micro-milling machine tool is one of the main sources of thermal errors. Other sources of thermal errors include drive elements like linear motors and bearings, the machining process itself and external thermal influences such as variation in ambient temperature. The basic strategy for alleviating the magnitude of these thermal errors can be achieved by thermal desensitization, control and compensation within the machine tool. This paper describes a spindle growth compensation scheme that aims towards reducing its thermally-induced machining errors. The implementation of this scheme is simple in nature and it can be easily and quickly executed in an industrial environment with minimal investment of manpower and component modifications. Initially a finite element analysis (FEA) is conducted on the spindle assembly. This FEA correlates the temperature rise, due to heating from the spindle bearings and the motor, to the resulting structural deformation. Additionally, the structural deformation of the spindle along with temperature change at its various critical points is experimentally obtained by a system of thermocouples and capacitance gages. The experimental values of the temperature changes and the structural deformation of the spindle qualitatively agree well with the results obtained by FEA. Consequently, a thermal displacement model of the high-speed micro-milling spindle is formulated from the previously obtained experimental results that effectively predict the spindle displacement under varying spindle speeds. The implementation of this model in the machine tool under investigation is expected to reduce its thermally induced spindle displacement by 80%, from 6 microns to less than 1 micron in a randomly generated test with varying spindle speeds. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Micro-milling Thermal compensation High-speed spindle Micro-machining Spindle growth model
1. Introduction Many fields ranging from bio-medical devices [1] to defense applications [2] are increasing using parts and devices with micron-scale characteristic dimensions. Miniature scale devices provide significant advantages over their macro-scale counterparts in terms of cost, size and performance [3]. Over the last few years these devices have tremendously grown in the complexity of their functions and level of performance which has directly impacted their design features, and has led to a need in increase of their manufacturing accuracy. To address this need for increased accuracy of manufacturing micro-scale parts and devices, machine tool builders have greatly improved the performance of their machining centers. However, dimensional errors arising due
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Corresponding author. E-mail address: [email protected] (D. Mukhopadhyay).
0890-6955/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2009.11.002
to temperature variations that distort various machine tool components, defined as thermal errors, account for a majority of loss in fabrication accuracy of thermally uncompensated machine tools [4]. Moreover, the feature sizes in micro-scale parts are of the same order as these thermal errors. Hence, the effects of these thermal errors are more pronounced in micro-scale machining. A spindle of a micro-milling machine tool is one of the main sources of thermal errors [5–7] along with drive elements such as linear motors and bearings, the machining process itself and external thermal influences like the variation in ambient temperature [7]. There are three primary methods for counteracting these thermal errors which can be categorized as: elimination or avoidance, resistance and control, and compensation tactics. Elimination or avoidance tactics tend to completely eliminate any change in dimensions due to thermal fluctuations. They are best (or rather can only be) implemented during the design phase of the machine tool. Some examples of these tactics are
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symmetric design and serpentine design of machine tool structures. Resistance and control tactics tend to reduce and/or oppose any change in dimensions due to thermal fluctuations. They are best implemented during the build phase of the machine tool. Some examples of these tactics are use of thermally insensitive materials and highly efficient components that generate very low heat in machine tool components. Compensation tactics tend to compensate for any change in dimensions due to thermal fluctuations. They can be implemented during any phase of the machine tool design and development. Some examples of these tactics include passive modeling of thermal deformation machine tool models and active thermal compensation using thermocouples and thermal-structural mapping functions in machine tool control software. The elimination and control tactics are generally preferred over the compensation tactics. However, under many circumstances implementation of a compensation scheme is required. These circumstances can include: improving the accuracy of a machine tool after it has been built and reducing the residual thermal errors after the implementation of elimination and control tactics. Extensive amount of research has been conducted towards developing various compensation tactics geared specifically towards combating the thermal error due to spindle displacement or deformation. Researchers have used multi-variant regression analysis [8–10], thermal error mode [11,12], artificial neural network [13], dynamic simulation [5] and others techniques to develop precise compensation schemes to counteract spindle deformation due to thermal effects. Some of these schemes are developed offline and then implemented on a machine tool as an internal spindle deformation model [9], where as others are implemented a real time basis [14,15]. In general, the majority of the work done previously has concentrated on using medium to large spindles to implement the developed compensation tactics. A micro-milling spindle differs significantly from a larger spindle. The differences are not only in relatively size and power consumption but also in maximum speeds and distribution of its power between the cutting process and the power needed to overcome the friction of a spindle’s bearings. A clear proportional relationship between the thermal growth of a spindle and its speed has been established by other researchers [8–13]. Since the maximum speed of a micro-milling spindle (upwards of 50,000 RPM) is generally higher than its larger counterpart (up to 20,000 RPM), this growth effect is expected to be relatively more significant for a high-speed micro-milling spindle. Additionally, in a larger spindle used in macro-machining most of the spindle power is used for cutting. However, in micro-machining only a small percentage of spindle’s power is used for cutting [15]; the rest is used to overcome the friction in its bearings. Hence, a significant portion of spindle thermal growth occurs during the heat generated in overcoming this friction. This portion of the spindle growth can be effectively counteracted by capturing the spindle deformation while running the spindle in free-air and developing a compensation scheme based on those growth values. Finally, the smaller spindles used for micro-milling are generally air cooled, since they do not have enough real-estate for installation of effective liquid-cooling channels. Since liquid cooling has a higher heat capacity compared to a similarly sized air-cooling channels, these micro-milling spindles may grow significantly even in the presence of cooling, thus necessitating a presence of a growth compensation scheme. Moreover, many of the small machine tools used for micromilling are used intermittently with significant number of starts and stops throughout the day. A implementation of a spindle displacement compensation scheme that not only predicts the steady-state growth at a specific speed but also the rate of growth
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(or the transient state of spindle growth) can lead to increased productivity by eliminating the ‘‘warm-up’’ time associated with many larger machines, in which the machine is warmed up for a specified amount of time and spindle speed to let the thermal growth stabilize before use. Hence, there is a clearly defined need for a high-speed micromilling spindle displacement compensation scheme that is relatively easy to implement within a short period of time, and is quite reliable in reducing the machining errors, that arise due to the thermal deformation of the micro-milling spindle. There have been many machine tools using medium to large spindles that use some sort of compensation scheme to counteract spindle growth. However, there is still a lack of use of spindle growth compensation schemes in many small machine tools that are geared towards micro-milling. This can be attributed to two main factors, the time and effort it takes to implement available compensation tactics on a machine tool in an industrial environment and the robustness of the implemented model [16]. This paper describes such a high-speed micro-milling spindle displacement compensation scheme that aims to reduce the spindle-associated thermal errors. The implementation of this scheme is simple in nature and it can be easily and quickly executed in an industrial environment with minimal investment of manpower and component modifications. This paper is organized as follows. Initially, a finite element analysis (FEA) is conducted on this spindle to determine the
Fig. 1. Schematic of spindle mounting; (a) isometric view and (b) side view showing the offset between the spindle centerline and the mounting points of its housing to the rest of the model (Note: the near-side linear motor and its coil are hidden in this view).
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temperature distribution of the spindle, motor and its housing. This temperature field is fed back into the FEA model to ascertain the resultant structural deformation at the tool-tip, which directly affects the accuracy of the micro-scale part that is fabricated by this spindle. The analysis results were compared against the experimentally measured temperatures of the setup and the deformation at the tool-tip at a specific speed. Additional experimental runs were conducted at various spindle speeds with setup temperatures and tool-tip deformations measured during each run. These experimental results were then extended to develop a micro-milling spindle displacement model and compensation scheme. Finally, this compensation scheme was applied to a random operating sequence of the spindle to demonstrate its effectiveness in reducing thermal errors due to the spindle deformation.
2. Finite element analysis A finite element analysis of the spindle growth under the heat generated by the motor and the bearings was conducted to get an approximate temperature distribution in the spindle, motor and its housing. The schematic of the setup (or the model) in which this analyses (and the later experimental tests) were conducted is shown in Fig. 1. In this setup, a NSK NR40-5100 high-speed micro-milling spindle with an EM3060 motor was used to develop the thermal compensation scheme (shown in Fig. 2). This spindle has two sets of ceramic ball bearings, one near the spindle nose and the other
near the back face of the spindle. Moreover the bearing near the spindle nose is 6.35 mm in width and the one near the back face of the spindle is 12.7 mm in width. The motor is attached to the spindle behind the rear set of bearings. Additionally, the motor itself has two sets of bearings, both 12.7 mm in width, one placed at the front end of the motor and the other at its back end. This spindle is held between its bearings by the clamp of the spindle housing. The spindle housing is then attached to the rest of the setup through its mounting points. Additionally, the motor is attached to the back of the spindle but is not supported on any other location. In order to approximately simulate the geometric thermal changes for the spindle, a brief static-thermal FEA of its structure was conducted. The spindle body was assumed to be made of solid steel. The base of the analysis setup is made of black granite and the rest of the structural components are made of steel. The FEA was conducted in ANSYS v11 (workbench module). The model was meshed into 9973 elements, both hexagonal and tetrahedral in shape (using the mixed meshing configuration). This level of meshing leads to a good tradeoff between the simulation accuracy and solution time. For gauging the temperature distribution in the model, heat-flux boundary conditions were applied to the major heat generating sources; the bearings (2.5e 3 W/mm2) and the motor of the spindle (5.5e 4 W/mm2). This bearing and motor heat load was calculated from the power drawn by the spindle while running freely at 50,000 RPM (approximately 17.5 W) and by assuming 80% efficiency for the spindle motor. Additionally, this spindle uses air-cooling channels through its body, to prevent the spindle from overheating when
Fig. 2. A NSK NR40-5100 high-speed micro-milling spindle with an EM3060 motor.
Fig. 3. (a) Temperature distribution of the analysis setup at 50,000 RPM; and (b) temperature distribution along the spindle (Note: spindle housing and clamp are hidden in this view).
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operating under high speeds. A forced air convection coefficient (of 1.5e 4 W/mm2) was applied along the spindle body [17] to simulate this cooling effect. Lastly, effect of heat dissipation from the entire setup was captured by applying a typical near still-air surface-to-air convection coefficient of 2.5e 5 W/mm2 [17] to all the setup surfaces exposed to air. Fig. 3 shows the result of the FEA including the temperature distribution in the spindle, motor and spindle housing under a spindle-operating speed of 50,000 RPM (while rotating freely in air). From Fig. 3(b) it can be deduced that spindle-motor junction has the highest temperature in the model, as this is the joining point of two main heat sources in this setup. As expected there is a slight temperature increase in the body of the motor. However, the spindle air cooling is effective in reducing the temperature of the spindle body as well as the front bearing. The rest of the model including the base, experience minimal temperature increases (approximately + 0.26 1C) due to the spindle and the motor heat conduction. This non-uniform elevated temperature distribution in the analysis model is consequently expected to result in tool-tip deformation that affects the accuracy of any small machine tool in which this spindle is used. To gage this tool-tip deformation, the previously obtained temperature distribution was fed back to a
Fig. 4. Deformation of the spindle, motor and housing due under thermal stresses.
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static-structural FEA setup on the same model. In this analysis, the reference temperature was set at 22 1C which was the same used in the static-thermal analysis. Additionally, the sides of the granite base (with normal along X axis) were structurally constrained in all directions. The initial ‘‘thermal-condition’’ of this analysis was specified to be the results of the previously conducted static-thermal analysis. Fig. 4 shows deformation of the model under this static-structural analysis with a thermal loading condition. From Fig. 4 it can be observed the tip of the spindle deforms by approximately 6.6 mm. This deformation is distributed both as an axial elongation, as well as a deformation along the positive Y direction. There is no net deformation along the X axis due to the symmetric design of the model about the Y–Z plane. The deformation along the positive Y direction is in the form of a tilt about the X axis. This is due to the temperature differences along the length of the spindle and the offset (in the Y-direction) between the center-line of the spindle and the mounting points of the housing [Fig. 1(b)].
3. Experimental setup The previously conducted FEA gave approximate values and locations of the expected high- and low-temperature zones, along with expected deformations at the tool tip. However, detailed experiments with the physical setup were conducted to develop a more accurate characterization of the thermal profile of the model and the resultant tool-tip deformations at various spindle speeds while rotating in free air. All these experimental tests used the same set-up as shown in Figs. 5 and 6. Several different sensors (including thermocouples and capacitive sensors) and data acquisition equipment were used in order to gather the data necessary for the spindle thermal characterization. The geometry of the spindle at the tool tip was measured axially through capacitive sensors (from Lion Precision). The temperature of the spindle and motor was measured through type T glass braided thermocouples (from Omega) at critical points. The actual spindle speed and the torque produced by the motor were fed back through the spindle motor controller in real time. Tables 1 and 2 show the specifications for the spindle, thermocouples, and capacitive sensors, respectively, that were used in the spindle characterization experiments. For these tests, thermocouple (TC) placements were determined by locating the thermal sources in the spindle and motor. The front of the spindle contains bearings that produce heat. One
Fig. 5. Schematic of the spindle thermal characterization experimental setup.
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thermocouple (TC #3) was placed at the top of the spindle to monitor this heat generation. The face of the spindle was in contact with the bearings and had two air vents used to cool the spindle. Two thermocouples were located in this area; one (TC #1) near an air vent, where the air would have the largest affect, and the other (TC #2) equidistance from the two air vents, where the air would have the smallest affect. Two thermo-
couples were placed at the connection between the spindle and the motor since heat is produced in driving the spindle; one thermocouple (TC #4) was placed at the back of the spindle and another (TC #5) at the front of the motor. The final thermocouple (TC #6) was placed at near the power supply at the back of the motor. See Fig. 7 for exact placement of the thermocouples. An additional thermocouple, (TC #7) was used to monitor the ambient temperature. The following tests were conducted to gage the tool-tip deformation under various spindle-operating speeds: 1. Run spindle at 10,000 rpm until temperature has stabilized (about 4 h). Stop spindle. 2. Force cool the spindle with compressed air, allowing it to return to previous starting temperature (about 1.5 h). 3. Repeat with spindle at 20,000, 30,000, 40,000, and 50,000 rpm. 4. Use data acquisition software to record data for the temperature and geometry of the spindle. Use CNC Data Acquisition program to record the spindle speed and torque during the test.
Fig. 6. Spindle thermal characterization experimental setup.
Table 1 Spindle specifications (NSK NR40-5100) for spindle thermal characterization. NSK NR40-5100 spindle specifications Maximum speed Air for Collet opening Spindle accuracy Weight Power consumption rating Maximum output power
50,000 RPM 0.55a–0.61 MPa o 2 mm [0.00008’’] 0.74 kg [1.63 bs] 1.8 A 350 W
Table 2 Thermocouple (Type T) and capacitive sensors specifications for spindle thermal characterization. Type T thermocouple specifications Tolerance A/D converter
1 1C or 0.75% High-resolution, 22-bit
Capacitive sensors specifications Bandwidth ( 3 dB) Peak-to-peak resolution RMS resolution Sensitivity
15,000 Hz 50 nm 10 nm 40 mV/mm
Fig. 8 shows the temperature reading from TC #1 for various operating speeds of the spindle for a single experimental run. Additionally, Fig. 9 shows the temperature readings from all the thermocouples at the maximum spindle speed of 50,000 RPM. The saw-tooth like variation in Fig. 8 and Fig. 9 was due to the cycling of the laboratory’s HVAC system. This can be clearly observed by noticing the cyclic variation (with a magnitude of 70.5 C) in the temperature values read by TC#7, which monitors the ambient temperature. The corresponding axial tool displacements as measured simultaneously by the capacitance sensors are shown in Fig. 10. From this graph it can be noticed that the time taken to reach the steady state is independent of the spindle speed and the magnitude of the steady-state values increases non-linearly with the spindle speed. The cyclic variations in the displacement values are due to the corresponding variations in the ambient temperature as mentioned earlier. The experimental spindle thermal characterization followed the finite element analysis, qualitatively both in terms of temperature distribution and the tool-tip deformation. The experimental data shows that under air-cutting conditions the highest temperature for the spindle was at the spindle motor junction (TC #4 and #5) due to the heat generated by the motor and lowest was at the front of the spindle (TC #1), mostly due to the convection cooling caused by the air flow through the spindle. This matched with the finite element analysis results [Fig. 3(b)] of the spindle body. Also, the data shows that at a spindle speed of 50,000 RPM, a typical micro-machining spindle speed, the axial tool displacement was approximately 5.3 mm. However, this displacement was simulated as 6.6 microns [Fig. 4]. The
Fig. 7. Placement of thermocouples for the spindle characterization tests.
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discrepancy between these two values can be attributed to underestimation of the convection cooling of the air flow through the spindle, assumption in the values of various fluxes used in the simulation and the lack of extensive knowledge about the internal structure of spindle. Through these tests, several key observations were made. As expected, the axial tool displacement follows the spindle temperature, which itself was affected by the ambient temperature. The ambient temperature for the tests fluctuated causing variations in both the spindle temperature and the axial tool displacement. The maximum axial tool displacement was close to 6 mm. The spindle temperature and the axial tool displacement data from the tests show an exponential rise to a constant steady state, a trend that matches the exponential spindle growth model in [18]. This steady state varies depending on the spindle speed but the exponential time constant does not. The exponential rise of the spindle to its steady state took about 3000 s.
27 10,000 rpm 20,000 rpm 30,000 rpm 40,000 rpm 50,000 rpm
26
Temperature (C)
25
24
23
22
21
0
0.5
1
1.5
2
391
2.5
Time (sec)
x 104
4. Model development
Fig. 8. Temperature readings for various spindle speeds.
30 thermocouple 1 thermocouple 2 thermocouple 3 thermocouple 4 thermocouple 5 thermocouple 6 thermocouple 7
29
Temperature (C)
28 27 26 25 24 23 22 21 0
0.5
1
1.5
2
2.5 x 104
Time (sec)
Fig. 9. Temperature readings for various thermocouples at 50,000 RPM spindle speed.
The results from the characterization tests of the spindle under different operating speeds was used to develop a spindle growth model that is exponential in nature, as shown in (1) and similar to ones reported in [14,18].
Dz ¼ z0 þ ðzss z0 Þð1 et=t Þ
ð1Þ
where, Dz is the change in spindle growth over a time t, z0 is the spindle deformation at time 0, zss is the steady state spindle deformation for a particular speed and t is the time constant for the spindle growth (which is different from the time constant when the spindle is contracting during its slow down to a lower speed). For this model, steady state and time constant values were found from the conducted experimental tests. This data is summarized in Table 3. The rise time and fall time data was averaged to calculate two time constants; one for warm-up and the other for cool-down of the spindle. The next step was to develop an equation to determine the steady-state axial tool displacements at different spindle speeds. The axial tool growth data from multiple experimental runs were graphed against the actual spindle speeds and fitted to a straight line (Fig. 11). The linear fit between the spindle-thermal displacement and the actual spindle rotation speed was a good fit (2), (R value of approximately 0.95) and allows for continuous determination of the steady state (zss in microns) at various spindle speeds
7 10,000 rpm 20,000 rpm 30,000 rpm 40,000 rpm 50,000 rpm
6 5
micron
4
5.34µm 3.70µm
3 2
2.03µm
1
1.15µm 0.97µm
0 -1
Table 3 Axial tool displacement exponential curve data. Commanded speed (rpm)
50,000 40,000 30,000 20,000 10,000
-2 0
0.5
1
1.5
2
time (sec) Fig. 10. Axial tool growth for various spindle speeds.
Warm-up
Cool-down
Rise time (sec)
Steady state (mm)
Fall time (sec)
Steady state (mm)
803.3 706.9 750 668 900
5.34 3.70 2.03 1.15 0.97
540 470 530 448.4 305.8
5.34 3.35 2.2 1.35 0.7
Warm-up time constant
Cool-down time constant
695.38 51.71
518.82 40.62
2.5 x 104
Average Std. Dev.
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in the direction away from the spindle body thus leaving the tooltip position unchanged. This is due to the clamping arrangement of the spindle and the location of the motor behind the spindle body in which the motor is free to expand without effecting the position of the tool tip.
6 y = 1.1083E-4x - 0.48942
Axial Displacement (µm)
5
R2 = 0.9464
4
5. Results
3 2 1 0 -1 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 x
Speed (rpm)
5
A random speed step test was designed to gage the effectiveness of the exponential spindle growth model thus developed. Fig. 13 shows the profile of the stepped speed test. Capacitance gages were used to measure the actual spindle deformation under the stepped speed profile test. The spindle growth model was then used to predict the spindle growth under the speed profile. Fig. 14 shows the effect of the growth model in compensating for the spindle displacement in the random speed test. In this figure, the blue line denotes the tool-tip deformation (axial) as collected on a real time basis and the green line denotes
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Fig. 11. Steady-state axial tool growth versus actual spindle speed.
4 Spindle Speed (rpm)
6
5 Axial Displacement (µm)
x 104
5
4
3
3
2
1
2
0
1
-1 0
1
2
3
4
5
6
7
8 x 104
time (sec) 0 0
5
10
15
20
Fig. 13. Stepped spindle speed test for testing the developed model.
25
Power (%) Fig. 12. Steady-state axial tool growth versus drawn spindle power.
6
(vspindle in RPM) that were not considered in the experimental runs. ð2Þ
Additional tests were conducted to verify if the axial growth mapped more accurately to power drawn by the spindle. For these tests, a brush with metal bristles was attached to the spindle instead of the tool to increase the torque produced by the spindle to rotate the brush at the same speed as the tool. This setup also captures the additional torque needed during an actual machining process. Fig. 12 shows the graph of axial growth vs. spindle power. It can be observed that there is no clear relationship between the spindle axial growth and the power drawn by the spindle. Hence, the axial growth of the spindle is more closely aligned with spindle speed than power. This observation can be explained as follows. When the torque needed by the spindle increases at a specific speed, the motor draws more power and consequently heats up. This additional heat generates a thermal load on the motor which overhangs from behind the spindle body. However, this thermal load is gaged to only elongate the body of the motor
Axial Thermal Growth (µm)
zss ¼ 1:11e4 vspindle 0:49
Speed Step Test Speed Step Model
5 4 3
6µm
2 1
1µm
0 -1 -2 0
1
2
3
4 Time (sec)
5
6
7
8 x 104
Fig. 14. Prediction of tool-tip deformation by the developed model.
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the predicted deformation of the tool-tip by the previously developed model. From Fig. 14 it can be observed that without implementing the spindle growth compensation scheme, the spindle growth can reach up to 6 mm. However, this growth can be reduced to 1 mm by implementing the developed compensation scheme (i.e. moving the Z-axis on a real-time basis by the value predicted by the model in the opposite direction), resulting in a reduction of 80% in errors due to thermal sources. Another point that can be noted from Fig. 14 is the initial jump in the tool-tip position. This can be attributed to the structural relaxation that the spindle bearings undergo when the spindle is initially switched on. Furthermore, this behavior is absent when the speed of the spindle is increased from a non-zero initial value.
6. Conclusions This paper presents a high-speed micro-milling spindle displacement compensation scheme that is based on the exponential growth model. The initial finite element analysis results match qualitatively with the experimentally observed growth data to form a simple compensation scheme that can be implemented in an industrial environment with minimal investment of time, material and personnel. The compensation scheme is able to reduce the effect of thermal errors by up to 80% under a random stepped speed test. To ensure that the above developed spindle thermal compensation scheme is robust under different application environments; several methods of compensation were additionally considered. These included using only the developed model, using only tool length checks using a laser sensor and using the developed model in conjunction with intermittent tool length checks. The first type of implementation, using only the developed model is less favored as compared to the last one as the error between the actual and the modeled axial tool displacement can increase once steady state is reached. This error can accumulate over time, leading to errors similar in magnitude to errors without implementing the compensation scheme. This error accumulation is the result of ambient temperature fluctuations and drift over time. Though ambient conditions will not be accounted for the model, allowing for intermittent tool length checks along with the use of the developed scheme will help to reduce these errors.
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Acknowledgements We thank the ONR, Office of Naval Research (Contract no N00014-08-C-0088) for funding a part of this research endeavor.
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