On-axis self-calibration of angle encoders

CIRP Annals - Manufacturing Technology 59 (2010) 529–534

Contents lists available at ScienceDirect

CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er. com/ci rp/ def a ult . asp

On-axis self-calibration of angle encoders X.-D. Lu *, R. Graetz, D. Amin-Shahidi, K. Smeds Precision Mechatronics Laboratory, The University of British Columbia, Vancouver, BC V6T 1Z4, Canada Submitted by W. Tyler Estler (1), Gaithersburg, USA.

A R T I C L E I N F O

A B S T R A C T

Keywords: Calibration Spindle Encoder

Angle encoder calibration is needed to improve the accuracy of rotary position feedback in manufacturing equipment. Although several self-checking methods have been developed, the fundamental problem of calibrating angle encoders remains unsolved. Existing methods require an encoder to be calibrated on specially designed angle comparators, but the process of transferring the encoder from the comparator to the encoder’s application axis typically introduces arc-sec level error. This paper presents the Timemeasurement Dynamic Reversal (TDR) method to calibrate encoders on their application axes. The TDR method has demonstrated 0.006 arc-sec uncertainty (2s), including the first 500 harmonic components. ß 2010 CIRP.

1. Introduction The best commercial angle encoders can achieve accuracy of down to 1 arc-sec with proper mounting. Higher-accuracy encoders are needed to measure rotational axis angular position in ultra-precision manufacturing and measurement applications, such as optical disk mastering, or maskless nanolithography [1]. In order to achieve accuracy well below 1 arc-sec, calibration is required to eliminate the repeatable error components. National metrology institutes have designed angle metrology systems with repeatability of a few thousandths of an arc-sec by incorporating stacked indexing tables for comparison calibration [2] or using multiple rotary encoders coaxially mounted on two rotary axes for cross-checking [3–8]. However, all the existing angle comparators cannot be used to calibrate angle encoders installed on manufacturing equipment without significant modifications to the machine. An encoder can be mounted on the angle comparator (calibration axis) to perform calibration, but the obtained error map will be useless when the encoder is moved to the manufacturing equipment (application axis), for the following reasons: (a) the encoder error can be significantly affected by eccentricity, scale deformation, and alignment errors introduced in installation process; (b) a major source of encoder error, the radial error motion of the application axis can be very different from that of the calibration axis; (c) the error map can change with conditions, such as working speeds, and temperature. Therefore, encoder error is not a geometric property associated with only the encoder itself, but is a characteristic influenced by the combination of the encoder, the attached axis of rotation, and the installation between them. Ideally, angle encoders should be calibrated on their application axes to ensure the calibration error map is consistent with the

* Corresponding author. 0007-8506/$ – see front matter ß 2010 CIRP. doi:10.1016/j.cirp.2010.03.127

encoder error in the application process and under operating conditions. However, most self-calibration methods developed for the above-mentioned angle comparators [2–8] are for specially designed stages usually with 4–16 encoder heads, 2 scales, and 2 concentric air bearings, and thus cannot be applied to general purpose angle encoders installed on the manufacturing equipment. A 5-head self-calibratable encoder [9] has been developed based on the Equal Division Averaged method (EDA) [4,5], which relies on cross-checking among multiple scanning heads. However, the achieved accuracy of the self-calibratable encoder is limited by the missing multiple 5th order encoder error harmonics components. In earlier work [10], a Time-measurement Dynamic Reversal (TDR) method was introduced to very quickly self-calibrate an encoder on its application axis at any rotating speed, providing all encoder error harmonics. However, uncertainties caused by limited timing resolution and assumption of free-response rotation were not addressed. This paper presents two key improvements to the TDR method, allowing higher accuracy and less sensitivity to time measurement resolution and air bearing intrinsic torque. 2. Improved TDR self-calibration method When an angle encoder is mounted on a precision rotation axis, Fig. 1 shows the output signals from a single encoder scanning head: two quadrature signals A and B with N zero-crossings per revolution and an index signal with one pulse per revolution. The zero-crossing locations in A and B mark spatial sampling events, which are labelled by k according to the happening sequence. The first sampling event (k = 1) is the first zero-crossing in signal A following the rising edge of the index signal. These spatial sampling events ideally are spaced with nominal spacing of, D0 = 1/N revolution. The actual spatial interval between the kth sampling event and the (k  1)th sampling event is labelled as Dk

X.-D. Lu et al. / CIRP Annals - Manufacturing Technology 59 (2010) 529–534

530

2

D1

3

6. 7 6 .. 7 6 7 6 7 6 DS 7 7; D¼6 6D 7 6 Sþ1 7 6 7 6 .. 7 4. 5

2

3 T Nþ1 6. 7 6 .. 7 6 7 6 7 6 T NþS 7 7; T ¼6 6T 7 6 Sþ1 7 6 7 6 .. 7 4. 5

DN 2

Fig. 1. Rotary encoder spatial and temporal sampling events.

(k = 1 to N), and the corresponding temporal interval is Tk, which can be easily measured with time-capturing electronics. Deviations in Dk from the nominal value D0 constitute encoder error. The goal of TDR self-calibration method is to derive Dk from the measured Tk, based on spindle free-response dynamics.

Here the basic TDR theory in [10] is summarized. As most precision rotational axes are supported by non-rolling element bearings, such as aerostatic bearings or magnetic bearings, the free slowing down dynamics (with rotary motor turned off) can be modelled as dv þ cv ¼ 0; dt

(1)

where v is spindle speed, t time, and c the normalized damping coefficient. Using a 2nd-order Taylor series expansion, the angular speed at the kth sampling event is represented as

vk ¼ v0 þ ak þ bk2

(2)

where v0 is the initial spindle speed, and a and b are damping coefficients to be estimated. From the time measurements, spindle speed is given as

Dk

(3)

Tk

and the spatial distance between each count is written as

Dk ¼ T k ðv0 þ ak þ bk2 Þ;

for k ¼ 1; 2; 3; . . .

(4)

Circular closure constrains the sum of all count intervals to be exactly one revolution: SþN X

Di ¼ N  D0

(5)

i¼Sþ1

where S is the start index of the summed data set. Combining Eqs. (4) and (5), the initial speed is found to be P

P

2 SþN SþN Ti  i ND i¼Sþ1 T i  i v0 ¼ PSþN 0  a Pi¼Sþ1 b P : SþN SþN i¼Sþ1

Ti

i¼Sþ1

Ti

i¼Sþ1

Ti

(6)

2

3 ðN þ 1Þ2 T Nþ1 6. 7 6. 7 6. 7 6 7 6 ðN þ SÞ2 T 7 NþS 7 6 2 k T ¼6 7 6 ðS þ 1Þ2 T Sþ1 7 6 7 6. 7 6. 7 4. 5 N2 T N

Eq. (8) is applied to two data sets with 1.5 revolution spacing between their starting locations:

! P P  N  D0 T i T  i2 2 P Pi T k ¼ Dk þ a P i  k T k þ b  k Tk Ti Ti Ti

(7)

where k = S + 1,   , S + N and S represents the summing operation from i = S + 1 to S + N. Reordering these N equations of (7) according to Dk index from k = 1 to N, (7) can be rewritten in vector form as m¼DþaU þbV

(8)

where P T i U ¼ P i  kT; Ti

V¼

P T  i2 2 Pi  k T; Ti

(9)

The difference of the two vector equations cancels the spatial interval vector D common to each data set and gives the following linear equation: 2 3 a1 6 b1 7 7 ½ U 1 V 1 U 2 V 2 6 (10) 4 a2 5 ¼ m1  m2 : b2 In early work [10], the damping estimation is based on the least-square fitting of Eq. (10) and the encoder error map is calculated with the estimated damping coefficients from Eq. (9). This damping estimation can minimize the estimation error for graduation vector D, but can possibly bring large error to the encoder error map, which is the cumulative sum of D. 2.2. TDR integration enhancement In order to eliminate the accumulated calibration error caused by the damping estimation, Eq. (10) is modified as follows, 2 3 a1 7 intð½ U 1 V 1 U 2 V 2 Þ 6 6 b1 7 ¼ intðm1  m2 Þ (11) 4 a2 5 w b2 where the int (.) operator represents a cumulative sum for each individual column vector in the operand. To minimize the error map calibration error, the damping parameters are estimated with least-square fitting as 2 3 a¯ 1 6 b¯ 1 7 6 7 ¼ ðW T WÞ1 W T ½intðm1  m2 Þ (12) 4 a¯ 2 5 ¯ b2 ¯ , are derived as The estimated spatial distances, D ¯ ¼ m1 þ a¯ 1 U 1 þ b¯ 1 V 1 ; D

Replacing the initial speed in (5) with (6),

N  D0 m¼ P T; Ti

TN

m1 ¼ D þ a1  U 1 þ b1  V 1 m2 ¼ D þ a2  U 2 þ b2  V 2

2.1. Basic TDR theory

vk ¼

3 ðN þ 1ÞT Nþ1 6. 7 6 .. 7 6 7 6 7 6 ðN þ SÞT NþS 7 6 7; kT ¼ 6 7 6 ðS þ 1ÞT Sþ1 7 6 7 6 .. 7 4. 5 NT N

(13)

and the encoder error map p(k) is then: pðkÞ ¼

k N X k X X ¯ i  D0 Þ  1 ¯ i  D0 Þ: ðD ðD N k¼1 i¼1 i¼1

(14)

2.3. TDR rotary vibration removal The above derivation is based on the assumption of spindle dynamics of Eq. (1). In reality, there may exist a small amount of disturbance torque between spindle rotor and stator caused by imperfect bearing surfaces. This torque can make the actual spindle rotor vibrate rotationally around the spindle axis. As a

X.-D. Lu et al. / CIRP Annals - Manufacturing Technology 59 (2010) 529–534

531

Fig. 2. The 4-head setup for removing rotary vibration.

result, the derived encoder error map from Eq. (14) will be the combination of encoder graduation error g(k), spindle radial error motion divided by scale radius r(k), and rotary vibration vðkÞ: pðkÞ ¼ gðkÞ þ rðkÞ þ vðkÞ:

(15)

The first two terms are the encoder errors to be calibrated, while the third term degrades the calibration accuracy. In order to remove vðkÞ from the calibration map (14), the following method can be applied for the 4-head setup in Fig. 2, with four scanning heads H1, H2, H3 and H4 evenly installed around the circular scale. Calibrated maps from each head are expressed as pH1 ðkÞ pH2 ðkÞ pH3 ðkÞ pH4 ðkÞ

¼ ¼ ¼ ¼

gðkÞ gðk  N=4Þ gðk  N=2Þ gðk  3N=4Þ

þ   þ

yðkÞ xðkÞ yðkÞ xðkÞ

þ þ þ þ

vðkÞ vðkÞ vðkÞ vðkÞ

N X

pH j ðk; vÞ  e2p jkn=N

(17)

From the Fourier coefficients of the four error maps in Eq. (16), the rotary vibrating motion Fourier coefficients V(n) can be extracted as  P H1 þ P H2 þ P H3 þ P H4 VðnÞ; n 6¼ 4m ¼ (18) GðnÞ þ VðnÞ; n ¼ 4m 4 where G(n) is Fourier coefficient of g(k). The four head average cannot extract rotary vibration harmonics multiple 4th order. For these harmonics, the transfer function from disturbance torque M(s) to measured rotary vibration motion vðsÞ can be represented as (19)

where J is rotary inertia. The disturbance torque generally depends only on the rotary position, and its excitation frequency will change with the spindle speed. Consequently, the induced rotary vibrating motion will be smaller at high speeds than that at low speeds. The encoder error map components g(k) and r(k) are much more insensitive to spindle speeds, typically changing only several thousandths of an arc-sec over 100 rpm. Based on this speed-dependent characteristic, G(n) + V(n) can be partitioned as GðnÞ þ VðnÞ ¼ GðnÞ þ

An

v2

:

20 X

VðnÞ  e2p jkn=N :

(22)

n¼20

Here only the first 20 harmonics are included, as typically the vibration harmonics beyond 20 are negligible. The vibrationremoved encoder error map for the jth scanning head is j ¼ 1; 2; 3; 4:

(23)

3. Simulation results The TDR self-calibration method is applied to a simulated spindle with 4 scanning heads. The simulation model includes encoder grating error, spindle radial error motion, and a disturbance rotary torque. Time stamps Tk are simulated at each encoder signal zero-crossing. The time measurement clock rate fc is simulated at 600 MHz, which is readily achievable in experiment. Three methods are applied to Tk: (a) TDR in [10], without integration enhancement or rotary vibration removal; (b) TDR with rotary vibration removal but without integration enhancement; (c) TDR with integration enhancement and rotary vibration removal. The calibration inaccuracy is defined as the difference between the true error map g(k) + r(k) and the calibrated encoder error map from the three methods. Fig. 3 shows a simulated result at 200 rpm. Without integration enhancement or rotary vibration removal, the inaccuracy is about 0.03 arc-sec rms. The vibration removal method improves calibration down to 0.01 arc-sec rms inaccuracy. When both integration enhancement and rotary vibration removal are implemented, the calibration inaccuracy reaches 0.002 arc-sec rms. Fig. 4 shows the rms inaccuracy results of three methods over a wide speed range. It is confirmed that

(20)

From the calibration maps pHj (k,v) taken over a wide speed range, the complex constants An can be identified and the complete rotary vibration Fourier coefficients are found as 8 P ðnÞ þ PH2 ðnÞ þ P H3 ðnÞ þ P H4 ðnÞ > < H1 ; n 6¼ 4m 4 (21) VðnÞ ¼ A > : n; n ¼ 4m 2

v

vðk; vÞ ¼

qH j ðkÞ ¼ pH j ðkÞ  vðkÞ;

k¼1

vðsÞ 1  MðsÞ Js2

The speed-dependent rotary vibration can be estimated as

(16)

where x(k) and y(k) are spindle radial error motions along the X and Y direction. At each spindle speed v, the nth Fourier coefficients pH j ðn; vÞ of the calibrated error map pH j ðk; vÞ can be calculated for the jth scanning head as P H j ðn; vÞ ¼

Fig. 3. Simulation calibration inaccuracy results at 200 rpm. The true encoder error map is shown in black (scale on the left); the inaccuracy curves (scale on the right) of (a) TDR without integration or vibration removal shown in green, (b) TDR with vibration removal only shown in red, and (c) TDR with both integration enhancement and rotary vibration removal shown in blue.

Fig. 4. Simulated calibration inaccuracy across speeds. Theoretical accuracy limit due to timing uncertainty shown in black; inaccuracy of (a) TDR without integration enhancement or vibration removal shown in green, (b) TDR with vibration removal only shown in red, and (c) TDR with both integration enhancement and rotary vibration removal shown in blue.

X.-D. Lu et al. / CIRP Annals - Manufacturing Technology 59 (2010) 529–534

532

Fig. 5. Experiment setup with four encoder scanning heads (H1, H2, H3 and H4) and two orthogonally mounted capacitance probes (P1 and P2).

Fig. 6. Typical self-calibration map at 200 rpm, with TDR integration enhancement without rotary vibration removal for each scanning head (H1, H2, H3 and H4).

rotary vibration and integration enhancement greatly improve the TDR method accuracy close to the theoretical limit set by the time measurement quantization:

s max ðvÞ ¼

v 2 fc

:

(24)

4. Experimental results Fig. 5 shows the experimental setup, consisting of an aerostaticbearing spindle (Professional Instruments 10R) and a bearing-less angle encoder (Heidenhain 4282C). Four scanning heads are installed on the spindle stator (H1, H2, H3 and H4), and the spacing alignment errors are adjusted within 2 arc-sec. The encoder drum scale has 32,768 grating periods. Correspondingly signals of each scanning head generate N = 131,072 zero-crossings per revolution. The whole spindle is motorized by a brushless motor. The encoder signals are measured with custom electronics at equivalent 600 MHz clock [11]. In addition, two capacitance probes (P1 and P2) are installed on the stator to read against the encoder drum for spindle radial error motion measurement. 4.1. TDR self-calibration experimental results Encoder signals were captured from 300 rpm down to 200 rpm. Fig. 6 shows typical TDR results for each of the four heads at 200 rpm, without rotary vibration removal. After removing the extracted rotary vibration, the vibration-removed encoder error map is shown in Fig. 7: the error maps for each scanning head are very similar to each other, except for 908 spatial phase shift between each head.

With Eq. (21), the identified spindle rotary vibration harmonic amplitudes are plotted versus spindle speed in Fig. 8. The linear slope of 40 dB/dec on each harmonic confirms that rotary vibrating motion during free-response is caused by a speedindependent disturbance torque exciting an inertial system. With both integration enhancement and vibration removal, the calibrated encoder error map harmonic amplitudes for head H2 are plotted versus spindle speed in Fig. 9. Each harmonic is observed to remain nearly constant amplitude with changes of only a few milli arc-sec. 4.2. Calibration result uncertainty The uncertainty of the calibration result is evaluated by comparing calibrated error maps (Eq. (23)) from 10 experimental tests at 200 rpm. The variance is calculated as, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 10 131072 X u1 X 1 2 s¼t ½qi ðkÞ  q0 ðkÞ 10 i¼1 131072 k¼1

(25)

where qi is the H2 error map from the ith test result and q0 is the average error map of all H2 test results: q0 ðkÞ ¼

10 1 X q ðkÞ: 10 i¼1 i

(26)

The calibration uncertainty (2s) for H2 is calculated to be 0.006 arc-sec, with only the first 500 harmonics are included. This uncertainty is believed to be caused by ambient temperature and pressure fluctuations uncontrolled in the environment.

X.-D. Lu et al. / CIRP Annals - Manufacturing Technology 59 (2010) 529–534

533

Fig. 7. Encoder error calibration results of each head (H1, H2, H3 and H4) at 200 rpm, with integration enhancement and vibration removal. Encoder error maps with full harmonic components (top) and filtered error maps with first 500 harmonics components (bottom).

4.3. Calibration accuracy verification by radial error motion Based on Eq. (16), the 2D spindle radial error motion harmonics [12] can be calculated from the error map Fourier coefficients as

XðnÞ þ jYðnÞ ¼

1 2½PH1 ðnÞ  P H3 ðnÞ j 1  2½PH2 ðnÞ  P H4 ðnÞ

measured directly by two capacitance probes (P1 and P2) in Fig. 5. The fundamental radial error motion X(1) + jY(1) measurement results from two methods are compared in Fig. 10. The difference between them is 1.1 nm rms, which corresponds to 0.0023 arc-sec on the 200 mm diameter scale. The main difference is believed to be the spindle radial error motion variation caused by temperature and supplied air pressure fluctuations. 4.4. Experimental comparison between TDR method and EDA method

where X(n) and Y(n) are the nth Fourier coefficients of spindle radial error motions x(k) and y(k) respectively. This provides a means for external validation of encoder error map calibration accuracy, by comparing the radial error motion measured from Eq. (27) and that

The EDA developed at NMIJ/AIST has been confirmed to be of the state of the art accuracy in several metrology institutes [6]. In our test setup, the EDA method for 4-head self-calibratable encoder [9] is implemented to compare with the presented TDR method. The EDA method does not rely on spindle dynamics and thus is insensitive to torque disturbance. However, when used as self-calibratable encoders [9], the EDA method cannot determine

Fig. 8. Extracted rotary vibration during spindle free slowing down process, vðk; vÞ for k = 1, 2, . . ., 6.

Fig. 9. Scanning head H2 error map calibration results over speeds, using TDR with integration enhancement and rotary vibration removal.

for n ¼ 1; 3; 7; . . .

(27)

X.-D. Lu et al. / CIRP Annals - Manufacturing Technology 59 (2010) 529–534

534

Table 1 Comparison of calibration error maps from two methods for head H2 at 200 rpm. EDA: 4 head self-calibratable encoder [9]; TDR: Time-measurement Dynamic Reversal with integration enhancement and vibration removal. The error map harmonic amplitudes are in milli arc-sec. n EDA TDR

1 1052.2 1053.2

2 323.8 323.7

3 227.4 226.9

4 N/A 97.6

5 74.3 74.4

6 175.6 175.5

7 42.0 42.1

n EDA TDR

8 N/A 18.4

9 8.6 8.5

10 8.9 8.9

11 21.9 21.8

12 N/A 31.6

13 9.9 9.9

14 0.3 0.3

n EDA TDR

15 9.9 10.0

16 N/A 8.9

17 9.8 9.9

18 12.0 11.9

19 9.2 9.1

20 N/A 9.7

21 8.0 7.8

5. Conclusion and outlook

Fig. 10. Spindle fundamental radial error motion derived from encoder calibration error map and from capacitance probes measurement.

An improved TDR technique for angle encoder self-calibration has been developed and experimentally tested. For this spindle dynamics-based calibration method, both integration enhancement and rotary vibration removal should be applied to achieve the highest accuracy, by minimizing the effect of time measurement resolution and small intrinsic torque inside air bearing spindles. The calibration experimental results demonstrated calibration uncertainties of 0.006 arc-sec (2s), when the first 500 harmonic components are included. A comparison experiment was performed between the TDR method and the EDA method [9] on the 4 scanning head setup, and the agreement was within 0.001 arc-sec for the first 21 harmonic components (excluding multiples of 4). In comparison with other self-calibration methods developed for dedicated angle comparators, the TDR method can be applied directly to angle encoders installed on application axis, eliminating errors introduced during the installation and alignment process. In addition, the TDR method achieves fast calibration of the entire error map harmonic components under arbitrary spindle working speeds. Acknowledgements

Fig. 11. Comparison of calibration error maps from head H2 at 200 rpm filtered with first 500 harmonic components. EDA: 4 head self-calibratable encoder [9], shown in red, and scale on the left; TDR: Time-measurement Dynamic Reversal with integration enhancement and vibration removal, shown in blue scale and scale on the left; the 4th harmonics from the TDR method but missed in EDA, shown in green and scale on the right.

encoder error map harmonics that are multiples of the number of scanning heads installed. Both methods are based on the time stamp of the zero-crossings of encoder signals. The signals from four encoder heads are time-stamped simultaneously during spindle slowing down around 200 rpm and the captured data are processed with both the EDA and the TDR methods. Table 1 shows a comparison of the first 21 harmonic amplitudes of encoder head H2 error maps derived from EDA and TDR methods performed on the same experimental time measurements. The error map results for each method are the average of 10 experimental tests at 200 rpm. The agreement between EDA and TDR results is within 0.001 arc-sec for all first 21 harmonic amplitudes, except those multiple of 4. The key advantage of TDR is that it can also determine the multiple 4th order encoder error map harmonics, which for this particular experimental setup compose approximately 7.6% of the amplitude of encoder error or 0.234 arc-sec peak-valley. Fig. 11 shows the encoder error of head H2 found from TDR and EDA calibration methods, as well as the error components of 4th order encoder error map harmonics found from TDR calibration that are missing in the EDA calibration.

This work is sponsored by KLA-Tencor, NSERC, and Canada Foundation for Innovation.

References [1] Srituravanich W, Pan L, Wang Y, Sun C, Bogy DB, Zhang X (2008) Flying Plasmonic Lens in the Near Field for High-Speed Nanolithography. Nature Nanotechnology 3:733–737. [2] Estler T, Queen H (1993) An Advanced Angle Metrology System. Annals of the CIRP 42/1:573–576. [3] Masuda T, Kajitani M (1989) An Automatic Calibration System for Angular Encoders. Precision Engineering 11/2:95–100. [4] Watanabe T, Fujimoto H, Nakayama K, Masuda T, Kajitani M (2001) Automatic High Precision Calibration System for Angle Encoder. Proceedings of SPIE 4401:267–274. [5] Watanabe T, Fujimoto H, Nakayama K, Masuda T, Kajitani M (2003) Automatic High Precision Calibration System for Angle Encoder (II). Proceedings of SPIE 5190:400–409. [6] Probst R (2008) Self-calibration of Divided Circles on the Basis of a Prime Factor Algorithm. Measurement Science and Technology 19:1–11. [7] Probst R, Wittekopf R, Krause M, Dangschat H, Ernst A (1998) The New PTB Angle Comparator. Measurement Science and Technology 9:1059–1066. [8] Just A, Krause M, Probst R, Bosse H, Haunerdinger H, Spaeth C, Metz G, Israel W (2009) Comparison of Angle Standards with the Aid of a High-Resolution Angle Encoder. Precision Engineering 33:530–533. [9] Watanabe T, Fujimoto H, Masuda T (2005) Self-calibratable Rotary Encoder. Journal of Physics Conference Series 13:240–245. [10] Lu X-D, Trumper DL (2007) Self-calibration of On-Axis Rotary Encoders. Annals of the CIRP 56/1:499–504. [11] Amin-Shahidi D (2009) Ultra-Precise On-Axis Encoder Self-Calibration for Fast Rotary Platforms, M.A.Sc. Thesis, UBC. [12] Lu X-D, Jamalian A, Graetz R (2009) A New Method for Characterizing Spindle Radial Error Motion. Proceedings of ASPE Annual Meeting .