Sin/cosine encoder interpolation methods: encoder to ...

Sin/cosine encoder interpolation methods: encoder to digital tracking converters for rate and position loop controllers Steven T. Jenkins*a and J. M. Hilkertb Texas Instruments, PO Box 65512, MS 352, Dallas, TX 75265 b Alpha Theta Technologies, PO Box 850367, Richardson, TX 75085 a

ABSTRACT Pointing and tracking applications usually require relative gimbal angles to be measured for reporting and controlling the line-of-sight angular position. Depending on the application, angular resolution and/or accuracy might jointly or independently determine the angle transducer requirements. In the past decade, encoders have been increasingly taking the place of inductive devices where the measurement of angles over a wide range is required. This is primarily due to the fact that encoders are now achieving very high resolution in smaller sizes than was previously possible. These advances in resolution are primarily due to improved encoder disk and detector technology along with developments in interpolation techniques. Measurement accuracy, on the other hand, is primarily determined by mounting and bearing eccentricity as it is with all angular measurement devices. For very demanding accuracy requirements, some type of calibration of the assembled system may be the only solution, in which case transducer repeatability is paramount. This paper describes a unique encoder-to-digital tracking converter concept for improving interpolation of optical encoders. The new method relies on Fraunhofer diffraction models to correct the non-ideal sin/cos outputs of the encoder detectors. Diffraction model concepts are used in the interpolation filters to predict the phase of non-ideal sin and cosine encoder outputs. The new method also minimizes many of the open loop pre-processing requirements and assumptions that limit interpolation accuracy and rate loop noise performance in ratiometric tracking converter designs. Keywords: ADC, DAC, data converters, encoders, rate feedback, control, interpolation, analog

1. INTRODUCTION Advances in encoder technology and manufacturing techniques over the past decade have resulted in impressive performance and reliability improvements along with size and cost reductions. As a result, linear and rotary type of encoders are presently used in more scientific, medical, industrial and military applications than perhaps any other motion measurement device and have displaced other technologies, such as resolvers, in many applications. While there are several types of encoder designs, the most common is the incremental optical encoder that senses the motion of a grated disc as it interrupts the path between a light source and a detector as shown in Figure 1. These devices can achieve arc-sec accuracies and resolutions in the millions of counts per revolution by integrating a precision disc with a high number of accurately spaced apertures, or gratings, and interpolating the detected light signal as it traverses each aperture. As in any measurement application, accuracy and resolution are independent design parameters and each can be important depending on the specific application. Whereas the accuracy of an encoder, similar to that of any measurement device, is dominated by mechanical considerations such as bearing, disc and mounting eccentricities, the resolution achieved, by the type of encoder considered here, is primarily determined by the characteristics of the light source, detector, disc grating and the interpolation algorithm employed [1-2]. Figure 2 is a plot of the angular error versus the encoder shaft angle for a typical precision encoder. The periodic sinusoidal error shown is primarily due to disc and bearing eccentricities, and the more random “hash” is caused by the light diffraction and detection process along with the particular interpolation algorithm in use. For a well designed encoder, the periodic mechanical errors tend to be repeatable and, if necessary, can be reduced somewhat by calibration against a master reference. The higher frequency random-like errors can not be improved by calibration, and usually determine the resolution and often the ultimate accuracy that can be achieved. In applications where rate is derived by differentiating this signal, there is an added incentive for understanding and improving the resolution and high frequency error [3]. Achieving higher resolution, while simultaneously reducing cost, requires an understanding of the encoder errors and the output of the true photo detector output versus angle. While the performance capability of encoder interpolation systems Acquisition, Tracking, Pointing, and Laser Systems Technologies XXII, edited by Steven L. Chodos, William E. Thompson, Proc. of SPIE Vol. 6971, 69710F, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.777741

Proc. of SPIE Vol. 6971 69710F-1 2008 SPIE Digital Library -- Subscriber Archive Copy

has been steadily improving and resolutions of millions of counts per revolution have been achieved, the costs of these improvements are of obvious concern. The primary purpose of this paper is to investigate the nature of this noise process and to propose an interpolation algorithm that could result in significant improvement in the achieved resolution of these devices without adding proportional cost increases. The advantage of such an algorithm is that it could potentially be applied to a wide variety of encoder devices via a relatively low cost DSP and / or FPGA processor. However, before discussing the details of this approach, we shall briefly review the work of other investigators in this area. Lepple [4] describes some of the dominate error sources limiting the accuracy of sinusoidal encoder data converters. Transmission line noise, synchronous sampling of sin and cosine, gain errors versus speed, phase errors between sin and cosine, and offset errors must all considered and minimized by careful selection of preprocessing and post processing algorithms. Burke [5] describes two main methods for data conversion. The arc-tangent method of data conversion offers advantages in minimizing data latency, however it requires matched phase filters to reduce noise. The second and more common method is the ratio metric method that uses heterodyning, cross-multiplication to form tracking loop error signals. Noise filtering and signal averaging are the primary advantages of the ratio metric tracking loops. Tracking loops and arc-tangent methods have Nyquist speed limitations. Sub-Nyquist methods reported by Buckner [5] can extend the upper speed limit by up to a factor of two. Traditional data conversion techniques will often require calibration to correct for the residual errors that were not corrected or accounted for in the original encoder model. When using lower cost encoders, specifications on duty cycle will force the system designer to abandon quadrature interpolation and use the leading edge of each channel to lower rate noise. Duty cycle errors, caused by distortions of the sin and cosine encoder signals, are one of the dominate error sources in interpolation and rate estimation. Duty cycle errors allowed by spec can be as high as 5% to 15%, and are in inherent in the slit diffraction process and optical disc manufacturing. The duty cycle errors will have both random and systematic components. Systematic errors include distortion of the ideal sin and cosine signals. They result in the need for post processing calibration methods reported by Watanabe [1].

Photodetector

tanS =

Light source

D

tanS sinS 0

D S

D

w

Ca nd 'ton or rl in i Fl tim

usinG

mV)

Fig.1. The fundamental light detection / diffraction process. When

a2 N to avoid saturation of the A/D, see Figure 9. Auto-ranging the analog gain, 2N, enables higher resolution for slow speed and zero speed applications.

Proc. of SPIE Vol. 6971 69710F-4

Encoder Signal Conditioning and State Estimation CAN Profi bus RS486

I s in

Ideal s in output

DSP / Microc ontroller FP GA

Quad. TLL A –A B–B

θ +w+ pi / 2

Icos = ∫

∆Σ DAC

θ + pi / 2

2000 X Interpolation I cos Ideal cos output

θ +w

Isin = ∫

∆Σ DAC

θ

F[ θ (t) ]

F[θ (t)]dt

ADC

F[θ (t)]dt

ADC

Digital processing Filtered “sin”

d I cos dt

+ I sin - I s in

Type II Anti-a lias ing Filtered “cos ”

d I sin dt

+I cos -I cos

Type II Anti-a lias ing

Analog Processing System Reference

Flas h Memo ry - Calibration - Custom code

1 0.9 0.8 0.7

I

I=F[θ]

0.6 0.5 0.4 0.3 0.2 0.1

Supply Voltage

0 -6

-4

-2

0

θ

2

4

6

Fig.4. Sin/cosine tracking converter “open loop” architecture with interpolation and state observer. This architecture requires higher resolution A/D conversion to achieve the same interpolation performance, which in turn forces slower sampling rates and higher cost to achieve results similar to the tracking loop configuration in Figure 3. The benefits of this architecture compared to the closed loop configuration come from the removal the D/A feedback path, to focus on DAC resolution and phase matching, and loosens the data latency and accuracy specs needed in the closed loop design. The drawbacks to this design are the loss of over sampling A/D speed provided by the architecture in Figure 3. The averaging needed to improve signal-to-noise on the tracking error is one of the limiting factors in achieving the desired interpolation performance, and is the reason that the A/D noise input sources must be minimized by integrating the A/D directly with the photodetector analog output.


3. MATHEMATICAL DESCRIPTION h ( x , t ) = V det_ out = (V max + η 1 ) ∫

x (t )+ w

x (t )

H ( x, t ) =

F ( x ) dt + η 2

d (h( x, t ) dx

(1) (2)

where , 2

⎛ sin( x ) ⎞ F ( x) = ⎜ ⎟ for x ≠ 0 ⎝ x ⎠ and , a [r + e1 sin( φ ( t )) ]n φ ( t ) x (t ) = λD

and

0 < f ( x ) < V max

Vdet_out

= photodetector output model, volts

H(x,t)

= partial derivative of output with respect to the phase state estimate, x

Vmax

= constant proportional to the light source intensity, volts

a

= slit width, cm

r

= disc radius, cm

e1

= disc eccentricity, cm

e2

= encoder mounting eccentricity, cm

D

= distance between the photo detector and the optical disc, cm

λ

= wavelength of light illuminating the slit, cm

φ(t)

= shaft angle, radians

t

= time, second

n

= number of lines per revolution on the optical disc

η1

= gain process noise from light source and photodetector

η2

= offset process noise due the detector (stray light noise), volts

w

= photo detector lens, aperture width, cm

(3)

(4)

Figure 2 shows the Fraunhofer diffraction curve, F(x), for x between -pi/2 and pi/2 and the modulation of the light available to the photodetector due to tilt and bearing run out that changes the spacing between the optical disc and the photo detector lens, D=d1 and d2. The lens modulation transfer function (MTF) also works to spread the light intensity along the x axis however, for this model the MTF is assumed to be 1 for simplicity. Any changes in the lens MTF can be approximated by DC offsets to in the actual disc to detector spacing. The results of integrating the detector output over x is shown in Figure 2, and shows how non-ideal sin or cosine output models are able to account for duty cycle errors inherent in most encoder designs. It follows, then, that the interpolation problem requires solving equation (1) for x(t) and dx/dt given constraints on the allowed d2x/dt2 in order to minimize the interpolation errors. To facilitate optimum noise rejection, this task is typically performed in a state observer. Various methods have been proposed by Balemi and Defililppis [6], and Buckner [7-8] to solve for “x” in the face the gain and offset uncertainty, however both assume that the sin and cosine functions are


perfect, and then add bias, phase, and gain parameter estimation to account for uncertainty and imperfections in the signal transmission path. Sinusoidal models result in model mismatch errors that require pre or post processing to complete the calibration and achieve the desired interpolation accuracy. 1 0.9

Legend: d1 = 0.747cm d2 = 0.670 cm

Normalized Intensity

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -6

-4

-2

0

2

Radians

4

6

Fig.5. Fraunhofer diffraction curves with detectors spacing increased by 11.5% resulting in more light collected for a given detector aperture, a= 3.09e-4 cm. 180

x

160

x+w

50%

140

Amplitude, mv

120 100

w

80 60 40 20 0 0

Ideal Theta, radians sin encoder Diffraction output sin encoder output 1

2

3

4

5

6

7

8

9

Radians Fig.6 Ideal Encoder sin models assume a 50% duty cycle. The actual encoder output duty cycle is typically < 50% and varies with the detector aperture width, w. Signal distortion on the encoder outputs limit interpolation accuracy.

Without correction, model mismatch between the actual encoder output and assumed encoder output will cause phase spatial frequency errors that are 2n, where n is the number of cycles per rev. Arc-tangent conversion methods normally rely on pre-processing to remove the non-ideal, Lissajous errors from the encoder inputs. Lepple [4] describes several ellipse fitting methods for correcting the input signals prior to interpolation. Ratiometric tracking converters with nonideal actual inputs from the photo detector will create a 4n spatial frequency in the residual error function, assuming signal distortion and gain mismatch that cannot be completely removed from the signal chain.


∧ 1 A sin( 4 n φ (t )) 2 ∧ ∧ 1 A sin( 2 n φ (t )) A cos( 2 n φ (t )) ≅ B sin( 4 n φ (t )) 2 ∧

A sin( 2 nφ (t )) A cos( 2 n φ (t )) ≅

≅

(5) (6)

∧ 1 ( A − B ) sin( 4 n φ (t )) 2

(7)

Matching the sin and cosine amplitudes minimizes the 4x error function created by heterodyning. The arc tangent method will have half the frequency because it does not multiply sin and cosine signals (see Figure 7a and 7b). All interpolation methods are also dominated by gain, phase, and offset errors, and the most common system approaches to remove those errors are described by Lepple [4]. The need to measure and correct the non-ideal Lissajous ellipse at all encoder speeds is the expected result of assuming ideal signals in arc-tangent and ratiometric conversion techniques. On-line calibration methods for correcting the gain and phase are reported by Balemi and Defilippis [6] for signals up to 500 KHz using 20 MHz sampling D/A sampling rates. Auto-calibration methods used by Balemi increase the system accuracy, but also increase the complexity and reduce the sampling frequency from 120 KHz to 80 KHz due to the additional 6 floating point multiplications and 6 additions needed in the gradient search algorithm. The additional software phase delays must be balanced against the system noise, dynamics, and feedback goals.

ithi ItnI pi©d [] Graph of error over one electrical cycle

Interpolation error over 5 fringes

0.3

5

Error, rad x 1e-6

Error, deg

0.2 0.1 0

-0.1 -0.2 -0.3 -0.4

.

3 1 -1 -3 -5 0

Angle, 1 sin cycle, 360 deg

1

2

3

4

5

Fringes

Fig.7a. Arc Tangent Data Conversion errors. J. Burke, J. Moynihan, and K Unterkofler.

Fig.7b..Interpolation errors. (courtesy Micro-e).

4. SYSTEM SIMULATION The block diagram in Figure 8 shows the open loop simulation model used in this report. The simulation ignored the dynamic phase errors from the anti-aliasing analog filter and gain errors across the filter were lumped with the A/D gain errors to simplify the analysis. Figure 10 shows the results of modifying the Extended Kalman filter, (EKF), used by Zimmerman [19] to solve for x, phase, between +/-pi/2, where the actual input comes from the Fraunhofer Diffraction model and the predicted output is an ideal sinusoid with no distortion. Figure 10c compares the phase tracking error outputs for the Fraunhofer diffraction model and the ideal sin encoder models. The interpolation phase errors obtained with uncorrected ideal sin models are significant compared to diffraction models which are used to define the actual encoder output. Closed loop simulations results are comparable depending on the resolution selected for the A/D and D/A (see Figure 9). The advantage of the closed loop design comes in the ability to add effective resolution and oversampling of the analog error. Also, because the A/D and D/A gain errors are inside the loop, the gain and bias algorithms can be used to null the tracking errors associated with the D/A feedback path.


Vmax(0)>Vmax

1 s

DSP AGC Algorithm

Vmax

Vmax [H ( x )]

⊗ VSin(t)

+ -

k ( s + a) s

1 s

A/D

[K ]

zk

+ -

x

+

k

[A ]

x

− k

+

Vmax [h(x )] Fig.8. Open Loop Block Diagram Extended Kalman Filter.

offset

DSP Bias Algorithm

Vmax

DSP AGC Algorithm

VSin(t)

k ( s + a) s

+ -

1 s

+

2

k ( z + 1) z −1

N

A/D

-

x

x+ w / 2 d ⎛⎜ ∫ F ( x)dt ⎞⎟ ⎝ x−w / 2 ⎠ dt

+

⊗

+

D/A

Figure 9: Closed Loop Block Diagram for Sin Phase Interpolation. Tracking error gain, 2N, enables higher resolution and sample rates for interpolation. M is the D/A bit resolution, 2M, and M>N to avoid A/D saturation.

2

Sin Encoder output ,volts

1.5 1 0.5 0 -0.5 -1 -1.5 -2 0

0.005

0.01

0.015

0.02

0.025

0.03

Time, seconds Fig.10a. Predicted output from the encoder “sin only” diffraction EKF model.


0.035

Sin Encoder Phase output, radians

1 .5

1

0 .5

0

-0 . 5

-1

-1 . 5

0

0 .0 0 5

0 .0 1

0 .0 1 5

0 .0 2

0 .0 2 5

0 .0 3

0 .0 3 5

Time, seconds

Fig.10b. Predicted phase output from the encoder “sin only” diffraction EKF model. Rectified Encoder phase error ,radians

0.15

Legend: Ideal sin model -----Diffraction model ____

0.1

0.05

0

-0.05

-0.1

-0.15 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Time, seconds

Fig.10c. Phase error from the encoder diffraction “sin only” EKF model and ideal sin model.

Figure 10c illustrates the magnitude of the tracking error and the difference between the ideal sin model and the diffraction model is a function of the signal distortions found in the encoder output. This simulation used a 100 KHz D/A sampling rate and a known signal amplitude, Vmax. Non-ideal sin/cosine outputs can be minimized in encoder manufacturing, or modeled and removed during data conversion using the Fraunhofer diffraction models with the appropriate physical parameters adjusted for the specific encoder. Tracking error transients occur at the minimum and maximum of the sin input and are expected as the input signal-to-noise increases. The Kalman gain increases at the transitions, but not enough to drive the error to zero in the presence of the expected noise. Since the cosine input is out of phase by pi/2 the cosine errors will be small during the sin minimum and maximums transitions. So, parallel EKF tracking filters used on each encoder channel would lead to two separate phase estimates, as shown in Figure 11a. Algorithms to blend the two separate phase estimates into a single phase output would be needed to remove the transient errors. Rectified Encoder Phase, radians

2

Sin phase

1.5 1 0.5 0 -0.5

Cosine phase

-1 -1.5 -2 0

0.002

0.004

0.006

0.008

0.01

0.012

Time, seconds

0.014

0.016

0.018

Fig.11a. Predicted Phase output (Rectified) from the encoder sin/cosine diffraction EKF model.


Sin – Cosine Encoder Phase errors ,radians

0.01 0.008 0.006 0.004 0.002 0 -0.002

Cosine Phase Error

-0.004 -0.006 -0.008 -0.01 0

0.002

0.004

0.006

Sin Phase Error 0.008

0.01

0.012

0.014

0.016

Time, seconds

Fig.11b. Phase error from sin EKF and cosine EKF encoder models.

Rectified Encoder output, radians

1.5

1

0.5

0

-0.5

-1

-1.5 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Time, seconds

Rectified Encoder phase error ,radians

Fig.12a. Phase output from rectified sin - cosine encoder inputs. 0.01 0.008 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 -0.008 -0.01 0

0.002

0.004

0.006

0.008

0.01

Time, seconds

0.012

0.014

0.016

Fig.12b. Phase error from Rectified sin - cosine encoder inputs.

Another alternative to minimize the tracking error transients is to track “rectified” sin and cosine channels in a single EKF with known and fix phase offsets. This allows for reductions in the transients and optimizes the tracking phase errors, see Figure12a. The results of running the diffraction model in a single EKF diffraction filter with rectified versions of the sin and cosine are shown in Figure 13. The sampled raw sin and cosine signal are split to create two new tracking signals Z1 and Z2. The new tracking signals insure that the phase output is between pi/2 and pi/4. The benefits of tracking Z1 and Z2 in a single EKF model are evident in the improved phase error transient tracking performance shown in Figure 12b.


Rectified Encoder inputs, volts

p 1.5

1

Z2 0.5

0

-0.5

Z1

-1

-1.5 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.016

0.018

0.016

0.018

Time, seconds Fig.13a. Predicted output (Rectified) from the encoder sin - cosine diffraction EKF model.

Rectified Encoder input (Z1), volts

p 0

-1*rectified sin input

cos input

-0.5

Z1 input used to make the 4N phase signal. -1

-1.5 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Time, seconds Fig.13b. Predicted output (Rectified) from the encoder sin - cosine diffraction EKF model. Rectified Encoder input (Z2), volts

1.5

1

0.5

0

Z2 input used to make the 4N phase signal.

-0.5

-1

-1.5 0

rectified sin input

rectified cos input

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Time, seconds

Fig.13c. Predicted output (Rectified) from the encoder sin/cosine diffraction EKF model.

In all algorithms the phase interpolation error is optimized when the gain of the predicted sin and cosine output matches the actual filtered encoder gain, Vdet_out. Figure 14 shows how the gain calibration controller minimizes the tracking measurement error by decreasing the predicted gain until it reaches minimum error tracking of the encoder input, Vdet_out. The initial condition of the predicted encoder output gain, Vmax, must be greater than the actual Vdet_out to allow the EKF filter to lock prior to starting the gain calibration controller.


-3

Measurement tracking error, volts

3

AGC algorithm: Tracking error residuals

x 10

2

1

0

-1

-2

-3 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Time, seconds

Fig.14. AGC algorithm used for on-line calibration of the tracking loop errors. The AGC loop works in parallel rather than serially to remove modeling errors and minimize phase associated phase tracking errors.

Table 1: Interpolation Error Budget

Parameter (T = A/D sampling period, 100 KHz.) Non-ideal sin encoder output V gain error [(Input freq *9)*(T/3)] -1 [(Input freq *3)*(T/3)] -1 A/D resolution (open loop) [(Input freq *3)*(T/9)] -1 A/D (N=0) + D/A resolution A/D (N=4) + D/A resolution w detector error w detector error w detector error w detector error [(Input freq)*(T/9)]-1 [(Input freq /9)*(T)]-1 A/D resolution (open loop) A/D (N=6) + D/A resolution

Model Error

Phase Error std, radians (1 electrical cycle = 2π)

1.6% duty cycle error 0.10% 312 samples/cycle 937 samples/cycle 10 bit 312 samples/cycle (0 + 10) bit + 10 bit (4 + 8) bit + 10 bit 0% -1% 1% -10% 937 samples/cycle 937 samples/cycle 16 bit (6 + 10) bit + 10 bit

5.00E-02 6.00E-03 3.70E-03 8.49E-04 6.68E-04 2.72E-04 2.53E-04 1.87E-04 1.22E-04 1.20E-04 1.18E-04 7.81E-05 5.30E-05 5.30E-05 3.92E-05 3.90E-05

Encoder disc deformation and axis tilt, see references [1], [20], [2], also need to be considered as they make D a function of φ. The dominate error sources listed in Table 1 need to be addressed and removed in any interpolation system. Other errors like, mounting eccentricity errors between the encoder shaft and the load shaft, are not observable in the encoder output and must be removed with calibration reference encoders. Additional photodetectors might also be required to aid in removing eccentricity errors if the errors cannot be modeled and removed in the state estimation filters.

5. CONCLUSION The open and closed loop interpolation algorithms presented in this paper allow for improved interpolation when using non-ideal encoder outputs. The tracking filters used are not based on the traditional assumptions typically used in sin/cosine ratiometric techniques. The new method uses diffraction models to define the encoder output and does not assume that the encoder signals are perfect sin or cosine. This allows one of the major sources of interpolation error to


be included inside the tracking filter, and therefore does not require algorithms to re-build encoder outputs into ideal sin and cosine functions. Open loop tracking converter techniques are limited by the cost, speed and precision of the A/D. The performance advantages in closed loop tracking configurations, compared to open loop designs, come from the use of high speed over sampling of the tracking errors. When using closed loop tracking, where the photodetector output is compared to the predicted diffraction model output, an 8 bit A/D converter with the error gain set to N=8 achieves the same tracking performance as a single 16 bit A/D, and allows for increased sample rates on high frequency encoder outputs. The ability to over sample the analog error signal allows for additional robustness and signal-to-noise improvement needed in high bandwidth interpolation filters. Integration of the A/D with the photodetector to achieve high bandwidth interpolation is recommended for optimal robustness and signal to noise reduction. Three key ideas are discussed in this paper. One, the DSP engine models the systematic signal distortion on the sin and cosine encoder signals and compares them with the non-ideal photodetector sin and cosine signals. Two, the “analog errors” signals are digitized so the DSP engine can estimate the phase and close digital tracking loops. The digital loops also minimize the systematic tracking errors associated with the gain and offsets on the encoder outputs. And three, the real encoder non-linear Fraunhofer diffraction intensity function and the photodetector integration window that describe a given encoder are modeled to allow increased interpolation and tracking accuracy. Further calibration and interpolation performance enhancements are planned to allow on-line calibration and parameter estimation of the nonlinear intensity function and the actual photodector integration window. Non-ideal sin/cosine encoder outputs can be minimized in manufacturing or modeled and removed during data conversion using the Fraunhofer diffraction models with the appropriate physical parameters adjusted for the specific encoder.

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15

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