Angular error correction

(a) (b)

Figure 5.39: FFT analysis of the signals of the AMR sensor analyzed in [47]. (a) FFT analysis of a well aligned sensor. (b) FFT of a sensor misaligned 1 mm away from the magnetic center. The main magnet rotational frequency is not exactly at 100 Hz. Due to this, higher harmonics are slightly shifted to right. Peaks at 0.5x and 1.5x fundamental are indeed the real rotational frequency of magnet what is doubled by AMR effect and in this data not completely suppressed what is usually the case. This effect is technology related and come e.g. due to process related biasing of xMR bridge element. Hence, the peaks at 0.5x and 1.5x are neglected in this investigation.

Based on [47], the following assumptions for harmonics distortion frequencies contained inx -andy-signal of the „fictional“ sensor pixel over the full rotation were applied:

Harmonic frequency: 2nd 3rd 5th 7th Amplitude distortion ratio [%] 0.10 0.07 0.05 0.08

Magnitude [dB] 60.00 63.09 66.02 61.93

Phase [] 0.00 0.00 0.00 0.00

Table 5.19: Assumed values for the magnitudes and phases of superimposed harmonics for the har-monics cancelation test. Amplitude distortion ratio is related to the initial amplitude of the ideal signal in time domain.

The maximum angular error, caused by applied harmonics amounts to 0.1132.

Figure 5.40 shows the angular error correction results obtained by homography approach in dependence of degrees of freedom and amount of equidistant distributed matching points.

3 4 5 6 8 9 10 12 15 18 20 24 30 36 40 45 60 72 90 120 180 360 Amount of matching points

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Angular error [°]

DOF = 6, 6 coeffs.

DOF = 8, 8 coeffs.

DOF = 8, 9 coeffs.

Figure 5.40: Investigation of linear angular error correction algorithm based on homography approach:

harmonics cancelation. Remained angular error by DOF 6 is the same as the initial one. Remained angular errors achieved by DOF 8 with 8 and 9 estimated coefficients are completely equally.

From figure 5.40 it is evident that this approach it is unable to decrease the angular error significantly. The remained error after homography based correction of DOF 6 equals to the initial angular one. Errors achieved by DOF 8 with 8 and 9 estimated coefficients are only barely better and completely equal. An interesting fact is that the angular errors at minimum required amounts of matching points, between 3 and 8, is higher than the initial one and only gets smaller if the amount of matching points is9. This arises due to unfavorable selection of points within the xy-paths for the error minimization.

Figure 5.41 shows the angular error correction results obtained by angular error correction algorithm based on geometric distortion approach in dependence of degree of the polynomial and amount of equidistant distributed matching points. The angular error reduction here looks different. With a polynomial degree of 2 the angular error is halved. A significant jump in the angular error correction tendency is achieved at polynomial degree of 4. Stability in angular degree reduction is obtained for all polynomial degrees at an amount of matching points around 60.

24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 136 140 144 148 152 156 160 164 168 172 176 180

Amount of matching points

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

Angular error [°]

Degree = 2 Degree = 4 Degree = 6

Figure 5.41: Investigation of nonlinear angular error correction algorithm based on geometric distortion compensation: harmonics cancelation. Also here the effect of unfavorable selection of points within the xy-paths in case of small amount of matching points is visible.

The comparisons of uncorrected and corrected angular error progressions for both approaches are in 5.42 and in 5.43.

By application of homography based angular error correction algorithms the angular error in Figure 5.42 gets pressed to the origin but the form of the progression is preserved. The max-imum angular error is reduced first by DOF 8. In the central area the error becomes slightly higher, but the maximum error stays in total smaller than the uncorrected one.

From Figure 5.43 it can be seen that the nonlinear error correction exhibits much better result but in the same time associated with discontinuity points, which impact limits the maximal an-gular error. However, their impact is significantly decreased by higher polynomial degrees. An extremely high reduction of remained maximum angular error is obtained by transition from 2nd to 4th polynomial degree.

To get the full knowledge about the performance of developed algorithms and obtain a direct comparison to 5.39 the corrected x^-/y-signals were analyzed in the frequency domain. The DFT analysis results for homography approach with DOF 8 are depicted in Figure 5.44, for nonlinear correction algorithm of degree 4 in Figure 5.45. The DFT analysis for all three degree variations for both algorithms is included in Appendix G.8.1.

0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 Encoder magnet angle [°]

-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Angular error [°]

DOF = 6, 6 coeffs.

DOF = 8, 8 coeffs.

DOF = 8, 9 coeffs.

Not corrected

X: 198 Y: -0.09453 X: 20

Y: -0.1132

Figure 5.42: Investigation of linear angular error correction algorithm based on homography approach:

harmonics cancelation. Comparison of angular errors before and after correction ( amount of matching points = 360). DOF 6 curve is overlapped by the uncorrected. Error progressions for DOF 8, 8 and 9 coefficients are completely equal. The maximum angular error is reduced from 0.1132to 0.0945.

0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 Encoder magnet angle [°]

-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Angular error [°]

Degree = 2 Degree = 4 Degree = 6 Not corrected

X: 20 Y: -0.1132

X: 360 Y: -0.05468

Figure 5.43: Investigation of nonlinear angular error correction algorithm based on geom. dist. com-pensation: harmonics cancelation. Comparison of angular errors before and after correction (amount of matching points = 181*4). Due to the algorithm functionality the angular error progression is split in 4 parts. High discontinuities occur which are minimized significantly by an increase of polynomial degree.

The maximum angular error is reduced from 0.1132to 0.05468by polynomial degree of 2, approx. to 0.02 by degree of 4 and approximately to 0.01 by degree of 6.

-400 -380 -360 -340 -320 -300 -280 -260 -240 -220 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0

Magnitude [dB]

Frequency components of the x-signal

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Frequency [Hz]

Ideal signal superimposed with harmonics Corrected signal

-400 -380 -360 -340 -320 -300 -280 -260 -240 -220 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0

Magnitude [dB]

Frequency components of the y-signal

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Frequency [Hz]

Ideal signal superimposed with harmonics Corrected signal

Figure 5.44: Investigation of linear angular error correction algorithm based on homography approach:

harmonics cancelation. Comparison of frequency components of the ideal signal superimposed with harmonic frequencies and corrected signal. By application of homography approach only the second harmonics is suppressed in this setup.Correction setup: DOF = 8, amount of matching points = 360.

-400 -380 -360 -340 -320 -300 -280 -260 -240 -220 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0

Magnitude [dB]

Frequency components of the x-signal

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Frequency [Hz]

Ideal signal superimposed with harmonics Corrected signal

-400 -380 -360 -340 -320 -300 -280 -260 -240 -220 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0

Magnitude [dB]

Frequency components of the y-signal

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Frequency [Hz]

Ideal signal superimposed with harmonics Corrected signal

Figure 5.45: Investigation of nonlinear angular error correction algorithm based on geometric distortion compensation: harmonics cancelation. Comparison of frequency components of the ideal signal super-imposed with harmonic frequencies and corrected signal. All supersuper-imposed frequencies are suppressed by this algorithm to a level < 100 db.Correction setup: Degree = 4, amount of matching points = 181*4.

According to the results of performed analysis it becomes clear that by application of homog-raphy approach with DOF 8 only the second harmonics is suppressed in this setup. All other harmonics remains. Due to suppression of the second harmonic additional group of harmonics arises in the signals which are below a level of 120 dB.

Due to the discontinuities in thex^-/y-signals by application of nonlinear algorithm a constant level of frequencies becomes visible in the spectrum. However, the occurred constant spectral level is below 90dB. This limit is under normal noise level in practice and therefore completely acceptable.

To measure the reduction of harmonics distortion in total and get the knowledge about signal quality increase, the THD described by the Formula 4.28 was calculated for both signal and for all three variations of degree over all components. The THDs are recorded in Table 5.20.

Not Homography Nonlinear correction

corrected DOF 6 DOF 8 DOF 8 2nd ord. 4th ord. 6th ord.

(6 coeffs.) ( 8 coeffs.) ( 9 coeffs.) (6 coeffs.) (8 coeffs.) (10 coeffs.) THD

x-signal 0.1297 0.1297 0.0987 0.0987 0.0370 0.0025 4.9781e-04 [%]

THD

y-signal 0.1297 0.1297 0.0987 0.0987 0.0530 0.0090 0.00433 [%]

Table 5.20: Angular error correction algorithms test: Harmonics cancelation. Comparision of THDs in % before and after angular error corrections.

The THD values of signals decrease after the homography based correction algorithm only barely. The THD values after the nonlinear correction fell to significant small values.

Conclusions: In general, the performed test shows that both methods are suitable for cance-lation of harmonics at the end of line calibrations. Using the angular error correction algorithm based on homography approach only modest results could be obtained. This method seems to be not powerful enough to get a significant cancelation of harmonics. A reduction of maxi-mum error by only approx. 16% is achieved. In comparison to it, the nonlinear error correction algorithm based on geometric distortion cancelation exhibits a much better result and fulfills the expectations. The polynomial degree of 4 seems to be sufficient. The maximum error due to harmonics interference with this polynomial degree is reduced by approx. 82%. Its caused constant noise level visible in the spectrum is below 90dB and thereby outside the practical relevance.

Compensation of misalignment effects (test 2)

This test represents the last test in the total investigations chain. The performance capability angular error correction algorithms for misalignment effects compensation is examined in this test. In accordance to the test plan, two encoder magnet shapes are investigated here in order to determine possible differences in the algorithm performance. Two misalignment conditions of encoder magnets were applied in line with the misalignment compensation investigations:

a) an extreme misalignment case and b) the worst SOA misalignment case determined in investigations in Section 5.7.2.

Extreme misalignment case

An extreme misalignment case was applied in this test to examine the aptitude of algorithms to compensate not only marginal mechanical errors but also major misalignments, what is still within the real conditions. The mechanical misalignment parameters of encoder magnet are shown in Table 5.21.

Misalignment x-translation y-translation z-distance x-tilt

case [mm] [mm] [mm] []

Extreme case -1.46 1.59 5 7

Table 5.21: Applied mechanical misalignment conditions of encoder magnets for the angular error cor-rection test: extreme misalignment case.

In Figure 5.46 the vector field and its components in case of applied misalignment of the disc magnet are illustrated.

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -7

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

(a) Field vectors (b) Bxcomponents (c) Bycomponents

Figure 5.46: Investigation of angular error correction algorithm: extreme misalignment case. Field com-ponents in applied misalignment case measured by the sensor array at encoder magnet angle = 90. The AMM point is outside of sensor array plane. Almostglobal gradientsare present in (a) and (b).

Nevertheless the curvature of the field components is still extractable. Simulation setup: 15 x 15 array, 1.59 mm x- and 1.46 mm y-translation, 7x-tilt, 5 mm z-distance, 14 bit ADC, disc encoder magnet.

Im Dokument Signal processing algorithms for magnetic sensor arrays (Seite 192-200)