CHAPTER 2. TWO-DOF ORIENTATION MEASUREMENT SYSTEM
2.3 Numerical Simulation and Experiment
2.3.1 Example 1: Two-DOF Orientation Measurement System with Single Sensor
Magnetic field of PM based on DMP
A cylindrical PM is rotating by and with a constant offset Lp. The DMP method is numerically simulated to compute MFD according to a rotation of the PM as shown in Figure. 2.12. Due to the symmetry of the magnetization of the PM along the z axis, only two Euler angles ( and in Figure 2.12) can be measured. The range of Euler angles is set as q={[ ]T: -14 ≤ and ≤ 14 degree} and dicretized with 1 degree interval. The total number of discretization is 841(=29×29) pairs by using the DMP method. The single sensor is located in the initial sensor position Pi = [0 0 Ls]T. Figure 2.13 shows the magnetic field distribution Bx, By, Bz, and B( BX2BY2BZ2)with respect to the orientation ( and ). The detailed parameters of the DMP method and the measurement system, which is used in simulation and experiment, is summarized in Table 2.1.
Figure 2.12 Coordinate systems of example 1
Figure 2.13 Magnetic field distribution
Table 2.1 Parameters of measurement system and DMP PM specifications and sensor/PM location
a = 6.5mm, l = 8mm, M = 1.19T, LP = 72mm, LS = 83mm DMP parameters
2
8, 2, 2.8 mm
mj 0.2354, 0.0478, 0.5015 T / m 1.0e 4 ( 0,1, 2)
n k l
j
Investigation of optimized sensor’s position and orientation
The Hall-effect sensor, the measurable MFD-field shape is square, was used in this system. The absolute maximum and minimum range of MFD on the each axis are same as
max,1 max,2 max,3 130mT
B B B and Bmin,1Bmin,2 Bmin,30.098mT respectively as the Hall-effect sensor of which the magnetic linear range is ±130mT with 0.098mT resolution was chosen for the experiment.
The unit sphere in Figure 2.9 can be applied in this system to identify the sensing performance because the two constraints in Eq (2.17) and (2.18) are satisfied. Figure 2.14 shows result of the sensing performance using a unit sphere when the sensor position Pi is set as [0 0 Ls=83]T. The area satisfying the both conditions simultaneously in Eq (2.14) and (2.15), AC12u, is in the unit sphere. The colored area is the area satisfying the both conditions and black sphere is the unit sphere. The sensing performance using unit sphere ηu is 3.2185%.
Figure 2.14 Sensing performance in unit sphere
(a) Ls = 81 (b) Ls = 82
(c) Ls = 83 (d) Ls = 84
3.2017% ( 81) 3.2017% ( 82) 3.2185% ( 83) 3.2152% ( 84) 3.2093% ( 85)
u s
u s
u s
u s
u s
L L L L L
(e) Ls = 85 (f) Sensing performance using unit sphere Figure 2.15 Sensing performance in a unit sphere according to Ls
Figure 2.15 shows sensing performance in a unit sphere according to Ls. The area AC12u is in orientation domain and the range of Ls is set as 80~85mm (Figure 2.15 (a) to (e)). In Figure 2.15 (a) and (b), the hollow area is appeared in the center of AC12u because the close distance between PM and sensor is occurring the sensor saturation. The sensing performance according to Ls is shown in Figure 2.15 (f).
The maximum sensing performance is appeared in Ls=83 because the sensor position not occurring the sensor saturation is Ls=83 and the portion where the signal is weak increases in the Ls=84 or more. As a result, the optimized sensor position is Ls=83 maximizing the sensing performance not considering the sensor’s orientation
(a) qs [45 45]T (b) qs [ 45 45]T
(c) qs[45 45]T (d) qs [ 45 45]T
Figure 2.16 The sensing performance in a unit sphere according to sensor’s orientation
Figure 2.16 shows sensing performance in a unit sphere according to the sensor’s orientation [ ]T
s s s
q . The area AC12u is in orientation domain and the qs is set as four cases (Figure 2.16 (a) to (d)). The shape is different and it has symmetric shape each other however sensing performance is same when the sensor rotates according to qs. It means that the sensor’s orientation is not an important factor affecting the sensing performance. In the measurement system of example 1, the sensor’s orientation is set as qs = [0 0]T. It means he sensor coordinate system is equal to the global coordinate system and it has the advantage of easy installation of the Hall-effect sensor in the experiment.
Analysis of bijective domain
Figure 2.17 shows the bijective domain using Jacobian matrix according to MFD components in orientation domain ((a) is using [Bx Bz]T, (b) is using [By Bz]T, (c) is using [Bx By]T, (d) is using [Bx By
(a) Using [Bx Bz]T
(b) Using [By Bz]T
(c) Using [Bx By]T
(d) Using [Bx By Bz]T and [Bx By B]T Figure 2.17 Bijection in orientation domain
In this system, the orientation domain is a plane because only two orientation angles can be measured.
The Jacobian matrix in Eq (2.7) is square matrix when a two MFD components are used. The solid line in Figure 2.17 is a contour of the determinant with its value. The area between zero determinants another is the bijective domain. Since there are two unknown orientation components (q=[]T), it requires at least two MFD components. However the area, which is satisfying the bijective relation using two MFD components, has singularity in = 0 ([Bx Bz]T) or = 0 ([By Bz]T) and there is a contour of zero determinant in the region q={[ ]T: -6 ≤ and ≤ 6 degree} ([Bx By]T). In the case using three MFD components ([Bx By Bz]T or [Bx By B]T in Figure. 2.17 (d)), the Jacobian matrix is non-square matrix as the dimension of B is bigger than q. The left inverse is considered and the determinant of (JTJ) can be calculated. The area which is using three MFD components has a simple circle shape of bijective domain than using two MFD components and has no singularity in the circle. B (=[Bx By B]T) are used for the ANN since the area satisfying the bijective relation between q and [Bx By B]T is larger than [Bx By Bz]T as shown in Figure 2.17 (d).
Figure 2.18 shows the MFD domain of B (=[Bx By B]T) and the narrow parts with orienatation domain.
Two conditions in Eq (2.5) and (2.6) are satisfied since B and q are uniquely determined without overlap as shown in Figure 2.18 (a). However, the results show B is not uniformly distributed unlike the q. As the PM moves around the conner of the square motion region in the orientation domain, the MFD becomes weak and the sensor can not measure the MFD in practice since a number of measurement of MFD are concentrated around the origin. The difficulty in measurement can be overcomed by
(a) MFD domain of B
(b) Narrow parts in MFD domain Figure 2.18 Orientation and MFD domain
Measurement accuracy based on Jacobian matrix
The Jacobian matrix consisting of gradient of B(=[Bx By B]T) with respect to q and the square root of a sum of its components are expressed in Eq (2.26) and (2.27). As the dimension of B is bigger than q, the Jacobian matrix is non-square matrix.
/ /
/ /
B / B /
x x
y y
B B
B B
J (2.26)
3 2
2 2
1 1
( , )
i j
G i j
J (2.27)Figure 2.19 shows magnitude of G according to and . Each color circle line is the contour of the same gradient and the value is the magnitude of G. The region inside the contour has higher gradient than that of the magnitude of G. The gradient in origin is the largest and as the distance from the origin increases, the gradient becomes small. As the gradient increases, the satisfied region decreases simultaneously as shown in Figure 2.19. The region with some or more magnitude of G can be approximated in the closed circle. The boundary condition is the radius (=magnitude of G) of outer circle.
Figure 2.19 Magnitude of G according to and
Measurement result by using ANN in simulation
The ANN for q is numerically simulated from B made by DMP method. For practical simulation, the
respectively and nb is set as only 0.1mT. The initial number of data is 841 pairs and the orientation is set as the square region (q={[ ]T: -14 ≤ and ≤ 14 degree}). As the interested region is closed circle, the edge of region does not have to be taken into account. Therefore, modified total number of data in closed circle for training the ANN is 625 pairs. Two layers with seven neurons is applied in the ANN (h=2, n=n1=n2=7) and the transfer function of each layers is given in Eq (2.28).
1 2
( ) ( )
n n
e e
f n f n en e n
(2.28)
where n is the number of neurons in ANN.
(a) n = +5%
(b) n = +10%
Figure 2.20 Measurement error according to rc by using ANN in simulation
Figure 2.20 shows simulated result using ANN with two MFD variation model. The maximum and mean error of and in each cloesd circle, which is approximated considering gradient in Figure 2.19, were shown in Figure 2.20, where rc 2 2
( ) is the radius of each circle. The error tends to converge to 0 when rc decreases and the error in origin is the smallest since it has the maximum gradient.
The simulation result shows that the measurement is more accurate in the large gradient.
Calibration for experiment
Figure 2.21 shows experimental setup of example 1 consisting of a cylindrical PM and single three- dimensional Hall-effect sensor (TLV493d). The Hall-effect sensor of which the magnetic linear range is ±130mT was chosen for the experiment. The Goniometer stage, which realizes rotation of α and β, was used as a reference. XYZ stage was used to set the position of sensor in a desired place.
Figure 2.21 Experimental setup for example 1
The Hall-effect sensor communicates MFD data using i2c. The resolution of 12-bit readout is 98uT and the sensor specifications related to the MFD are summarized in Table 2.2.
Table 2.2 Hall-effect Sensor specifications
Magnetic linear range ±130mT
Resolution 12-bit readout 98uT/LSB
Magnetic noise 0.1mT
Resolution drift ±20%
Offset drift -1 ~ +1mT
Due to manufacturing errors and imperfection of experimental setup, the positions of the actual measuring point and the designed measuring point can differ. Considering this issue, the measuring points of sensor are calibrated by using XYZ stage shown in Figure 2.22 (a) before applying the Hall- effect sensor to the measurement system. The XYZ stage can move in 3 DOF with ranges of 20+mm in two planar directions and 10mm in a vertical direction with 10μm travel resolution. Due to the symmetry of the magnetization of the PM along the z-axis, the actual position of the measuring points can be found by moving stage. It is possible to find the center of yz plane by looking for that the value of By and Bz are zero simultaneously as shown in Figure 2.22 (b). Similarly, the center of xz plane can be found by looking for that the value of Bx are Bz are zero simultaneously as shown in Figure 2.22 (c).
Table 2.3 summarizes the calibration results of the sensor’s measuring points.
(a) Setup for the calibration of sensor’s measuring point
(b) Magnetic field and yz plane of sensor (c) Magnetic field and xz plane of sensor Figure 2.22 Calibration of sensor’s measuring points
Table 2.3 Calibration results of the sensor’s measuring points in example 1 (unit: mm) Position of Measuring points (desired + discrepancy)
X axis Y axis Z axis
0 + 1.015 0 + 0.31 0 - 0.34
The Goniometer stage, TP 65-W30-W40, was used as a reference of orientation as shown in Figure 2.23 [23]. It can rotate orientation angles with 200mdegree travel resolution and the center of rotation is located 20mm above the mounting surface. The Goniometer enables a precise adjustment with the help of precision worm gears and adjustable dovetail guide. However, the actual adjustment by worm gears can differ from a defined scale mark due to a gear elasticity and manufacturing error. Considering this issue, the adjustments of Goniometer stage was calibrated to find actual adjustment.
Figure 2.23 Goniometer stage
(a) α adjustment range (b) β adjustment range
Figure 2.24 Adjustment range of Goniometer stage
Maximum adjustment ranges of each stage are ±15° and ±20° respectively indicated in product information as shown in Figure 2.24. However, actually it can move ±17° and ±23° respectively. When each stages moved from minimum to maximum ranges, the scale mark of front changes. In α adjustment range, the adjustment range per rev of warm gear is 1.6°. The 21 rev + 0.4° is required to move α stage from minimum to maximum range (1.6°×21+0.4°=34°). However, the 20 rev + 1° is required in the actual result (1.6°×20+1°=33°). It means the actual unit degree is 33°/34° = 0.9706°. Similarly, the β adjustment also calibrated. In β adjustment range, the adjustment range per rev of warm gear is 2.8°.
The actual unit degree is 44.8°/46° = 0.9739° because the difference is (2.8°×16+1.2°=46°) and
(2.8°×16=44.8°). Table 2.4 shows each stage’s scale mark of front indicating α and β orientation angles.
Table 2.4 Scale mark of front indicating α and β orientation angles in Goniometer stage
Stage 1 (=α) Stage 2 (=β)
Orientation angle The scale mark of front Orientation angle The scale mark of front
-17 0.8 -23 1.25
-16 0.170588 -22 2.223913
-15 1.141176 -21 0.397826
-14 0.511765 -20 1.371739
-13 1.482353 -19 2.345652
-12 0.852941 -18 0.519565
-11 0.223529 -17 1.493478
-10 1.194118 -16 2.467391
-9 0.564706 -15 0.641304
-8 1.535294 -14 1.615217
-7 0.905882 -13 2.58913
-6 0.276471 -12 0.763043
-5 1.247059 -11 1.736957
-4 0.617647 -10 2.71087
-3 1.588235 -9 0.884783
-2 0.9588242 -8 1.858696
-1 0.329412 -7 0.032609
0 1.3 -6 1.006522
1 0.670588 -5 1.980435
2 0.041176 -4 0.154348
3 1.011765 -3 1.128261
4 0.382353 -2 2.102174
5 1.352941 -1 0.276087
6 0.723529 0 1.25
7 0.094118 1 2.223913
8 1.064706 2 0.397826
9 0.435294 3 1.371739
10 1.405882 4 2.345652
11 0.776471 5 0.519565
13 1.117647 7 2.467391
14 0.488235 8 0.641304
15 1.458824 9 1.615217
16 0.829412 10 2.58913
17 0.2 11 0.763043
12 1.736957
13 2.71087
14 0.884783
15 1.858696
16 0.032609
17 1.006522
18 1.980435
19 0.154348
20 1.128261
21 2.102174
22 0.276087
23 1.25
The weight and bias of each axis (a1 and a2) for matching the simulated MFD based on DMP to actual measurements of sensor were determined by minimizing the error in Eq (2.18). It requires one or more actual measurements that are not zero value in each axis. The first column of Figure 2.25 shows the weight and bias of each axis according to the number of actual measurements Ns and the second column of Figure 2.25 is the calibration error of MFD between simulated MFD and actual measurements.
(a) On the x-axis
(b) On the y-axis
(c) On the z-axis
Figure 2.25 Weight and bias of each axis according to the number of actual measurements
The weight and bias of each axis is almost same when Ns is 10 or more. However, if the Ns is less than 10, the weight and bias have large difference from the value of steady state. Although 10 actual measurements are enough to find weight and bias, a fewer calibration error can be obtained if 20 or more actual measurements are applied. Measuring 10 and 20 actual measurements, the weight and bias were determined and summarized in Table 2.5 with calibration error. Figure 2.26 and 2.27 shows the MFD calibration of 625 data applying weight and bias with Ns=10 and 20 respectively (625 data is the total number of data in rc=14 (deg) closed circle).
Table 2.5 Weight, bias and calibration error of each axis Ns
Measuring
axis Weight Bias Maximum
Error (mT)
Mean Error (mT)
20
x -1.18268 0.204026 4.059862 1.079803
y 1.178557 0.054248 3.509832 1.044628
z 1.035189 -1.41081 3.215983 1.031387
x -1.18131 -0.17279 4.490315 1.130565
(a) MFD on the x-axis
(b) MFD on the y-axis
(c) MFD on the z-axis
Figure 2.26 MFD calibration of 625 data applying weight and bias with Ns=10
(a) MFD on the x-axis
(b) MFD on the y-axis
(c) MFD on the z-axis
Figure 2.27 MFD calibration of 625 data applying weight and bias with Ns=20
where the red and blue solid line in first column is actual measurements and calibrated MFD by applying weight and bias with Ns=10 (Figure 2.26) and 20 (Figure 2.27) respectively; second column shows the calibration error of each axis.
Experiment result
by putting the MFD of Hall-effect sensor into the trained ANN, which is trained by calibrated MFD, and compared with a reference by using Goniometer stage.
Figure 2.28 shows experiment result using ANN. The maximum and mean error of and in each circle, which is approximated closed circle in simulation, were shown and the numerical values are summarized in Table 2.6. The experiment result is similar to simulation, i.e. the error tends to converge to 0 when rc decreases and the error in origin is the smallest. It means the measurement is more accurate in the large gradient as confirmed by simulation.
Figure 2.28. Measurement error according to rc by using ANN in experiment
Table 2.6 Numerical value of experiment measurement error (unit: degree) rc
Mean Max Mean Max
14 0.2130 0.6823 0.2052 0.6678
13 0.2411 0.6640 0.1659 0.5738
12 0.1706 0.5457 0.1580 0.4660
11 0.1692 0.4419 0.1452 0.3989
10 0.1389 0.4174 0.1292 0.4100
9 0.1210 0.3659 0.1108 0.3134
8 0.1599 0.3678 0.0955 0.2604
7 0.1238 0.2844 0.0897 0.2100
6 0.0841 0.2408 0.0652 0.1683
5 0.0758 0.2204 0.0845 0.1926
4 0.0465 0.1457 0.0447 0.1714
3 0.0362 0.1079 0.0618 0.2370
2 0.0476 0.0720 0.0532 0.0840
1 0.0509 0.0610 0.0506 0.0650
q=[0 0]T 0.05368 0.0481