CHAPTER 2. TWO-DOF ORIENTATION MEASUREMENT SYSTEM
2.3 Numerical Simulation and Experiment
2.3.2 Example 2: Two-DOF Orientation Measurement System with Multi-Sensor
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
respectively. Table 2.7 shows rotation angles of four sensors.
Figure 2.29 Coordinate systems of example 2
Table 2.7 Rotation angles of four sensors (unit : degree) qs1=[10 -10]T, qs2=[-10 -10]T, qs3=[10 10]T, qs4=[-10 10]T
The magnetic field distribution discretized by DMP of each sensor is same in Figure 2.13 but the orientation region differs. The center of orientation region in Figure 2.13 was set as qs=[0 0]T because the magnetic sensor of example 1 is in the initial sensor position. It means the center of orientation region is the rotating angles of each sensor. Therefore, the center of orientation region for each sensor is the sensor’s rotating angles.
Multi-sensor approach
The issues related to the orientation measurement system are studied and simulated in the chapter 2.3.1 Example 1. The optimized sensor’ position is Ls=83 maximizing the sensing performance.
Sensor’s orientations are same as global coordinate system because it is easy to install the Hall-effect sensors in the experiment. Bijective domain is analyzed and B (=[Bx By B]T) is selected as MFD vector.
The measurement accuracy is considered using Jacobian matrix. The gradient in origin is the largest and becomes small as the distance from the origin increases. The region with boundary condition can be approximated as closed circle and measurement is more accurate in the large gradient. With this characteristic of bijective domain, the measurement error of each closed circle could be identified in Figure 2.28 and Table 2.6. Based on this result, the measurement system with desired accuracy,
maximum and mean measurement error, can be designed to expand a sensing region by connecting the closed circle having desired accuracy.
(a) A closed square in orientation domain (b) Range expansion using a closed square Figure 2.30 Region expansion using a closed square
The closed circle can be approximated to a closed square with a length ac=rc×cos45°×2 of one side as shown in Figure 2.30 (a). Closed Square having their motion region can be connected to each other easily in orientation domain. Thus a closed square with desired accuracy can be used to expand a sensing region as shown in Figure 2.30 (b). Four closed square are connected each other and the expanded motion region have a length 2×ac of one side. The expanded range of closed square according to the number of sensors is organized as Table 2.8.
Table 2.8 Expanded range of closed square according to the number of sensors The number of sensors Expanded range of closed square with a desired accuracy
1 qc={[cc ]T: -ac/2 ≤ c and c ≤ ac/2 degree}
4 qc={[cc ]T: -2×ac/2 ≤ c and c ≤ 2×ac/2 degree}
9 qc={[cc ]T: -3×ac/2 ≤ c and c ≤ 3×ac/2 degree}
...
m2 qc={[cc ]T: -m×ac/2 ≤ c and c ≤ m×ac/2 degree}
In the example2, the desired square motion region is set as q={[ ]T: -19 ≤ and ≤ 19 degree}.
Based on the result of single sensor in Table 2.6, the number of sensors can be determined to measure the desired motion region with desired accuracy. The expanded range of square according to the number of sensors is summarized in Table 2.9.
Table 2.9 The number of sensors with desired accuracy (unit: degree)
rc ac/2
The number of sensors
4 9 16 25
Expanded range of square according to the number of sensors
14 9.90 19.80 29.70 39.60 49.50
13 9.19 18.38 27.58 36.77 45.96
12 8.49 16.97 25.46 33.94 42.43
11 7.78 15.56 23.33 31.11 38.89
10 7.07 14.14 21.21 28.28 35.36
9 6.36 12.73 19.09 25.46 31.82
8 5.66 11.31 16.97 22.63 28.28
7 4.95 9.90 14.85 19.80 24.75
6 4.24 8.49 12.73 16.97 21.21
5 3.54 7.07 10.61 14.14 17.68
4 2.83 5.66 8.49 11.31 14.14
3 2.12 4.24 6.36 8.49 10.61
2 1.41 2.83 4.24 5.66 7.07
1 0.71 1.41 2.12 2.83 3.54
where the first column is a radius of each closed circle; the second column is half length of one side of each closed square; the desired accuracy is in Table 2.6 according to the radius of each closed circle.
The desired motion region can be measured using 4 sensors with the measurement accuracy of the rc=14 closed circle as shown in Table 2.9. The 9, 16 and 25 sensors can be used to measure the desired motion region with the measurement accuracy of the rc=9,7 and 6 closed circle respectively. It means more sensors are used, better accuracy can be obtained. The number of sensors is square of m as shown in the first column of Table 2.8. When m is odd number, the sensors are connected to each other based on the sensor where the position is in the initial sensor position. On the contrary, when m is oven number, the sensors are connected to each other without the sensor where the position is in the initial sensor position as shown in Figure 2.30 (b).
After the sensor's positions are determined, the switching condition is required to distinguish which sensor is used. The motion region to measure with MFD of ith sensor (=ΩSi) can be defined using closed circle. The ΩSi can be defined by using magnitude of MFD in Eq (2.29) because the magnitude of MFD is also circle shape as shown in Figure 2.13 and the boundary condition can be defined by the magnitude of MFD.
Si: Bi i
(2.29)
where Bi is the magnitude of MFD on ith sensor; εi is the boundary condition of each circle.
Figure 2.31 shows region expansion using multi-sensor with switching conditions.
Figure 2.31 Region expansion using multi-sensor with switching conditions
were εi is the magnitude of MFD in each closed circle and it means boundary condition of each circle.
When the motion region is connected to each other, the overlapped areas exist and there are many solutions as mentioned in chapter 2.2.6. As all solutions in the overlapped area are satisfying the bijective, every solution can be applied to obtain orientation from MFD of each sensor. Therefore, the sensor having a biggest gradient of B with respect to q is used because it has high accuracy.
Experiment
Figure 2.32 shows an experimental setup consisting of a cylindrical PM and four three-dimensional Hall-effect sensors. The model of the Hall-effect sensor is TLV49d used in the experiment of example 1 and the Goniometer stage is also same. The Hall-effect sensor located in the initial sensor position is used to set the sensors in a desired place.
Figure 2.32 Experiment setup for example 2
The sensor’s measuring points are calibrated by using XYZ stage implemented before in Figure 2.22.
Table 2.10 summarizes the results.
Table 2.10 Calibration results of multi-sensor’s measuring points in example 2 (unit: mm) Position of Measuring points (desired + discrepancy)
Index of sensors X axis Y axis Z axis
1 +0.955 +0.245 -0.315
2 +1.03 +0.32 -0.343
3 +1.035 +0.26 -0.303
4 +1.01 +0.225 -0.363
10 or more actual measurements per sensor are required to obtain the weight and bias minimizing the error in Eq (2.18). The weight and bias of each axis according to the sensor were determined by applying the 20 actual measurements and summarized in Table 2.11 with calibration error. 81data of MFD were measured using Goniometer stage and Hall-effect sensor to evaluate measurement performance. Figure 2.33 shows the MFD calibration 81data applying weight and bias of each sensors. The calibration error is in the Figure 2.34. The ith column of Figure means the MFD calibration and error on the ith axis (i=1,2,3 represent x,y,z axis) and the jth row of Figure means the MFD calibration and error of each sensor (j=1,2,…,4 represent index of sensor). The red and blue solid line in Figure 2.33 are the actual measurements and calibrated MFD by applying weight and bias respectively.
Table 2.11 Weight, bias and calibration error of each axis for multi-sensor Index of
sensor Ns
Measuring
axis Weight Bias Maximum
Error (mT)
Mean Error (mT)
1
20
x 1.1221 -0.2113 2.7381 0.4469
y 1.1214 -0.0819 5.0372 0.5338
z 0.9651 0.1057 1.4518 0.3009
2
x 1.2386 -0.1733 4.4158 0.6624
y 1.2510 -0.0658 2.8450 0.3581
z 1.0593 -0.0386 5.3849 0.6494
3
x 1.1377 0.2544 3.4524 0.5071
y 1.1441 -0.1558 3.8422 0.3456
z 0.9995 0.1368 1.8687 0.2565
4
x 1.1199 0.3266 3.6256 0.6328
y 1.1332 0.0740 2.8603 0.4538
z 0.9199 0.1959 6.4288 0.8467
Figure 2.34 MFD calibration error of 81 data applying weight and bias of each sensors
The ANN of each sensor are trained by using the calibrated MFD of each sensor. After training, the q was computed by putting the MFD of ith Hall-effect sensor (in the ΩSi) into the trained ith ANN. The initial desired motion region is set as q={[ ]T: -19 ≤ and ≤ 19 degree}. Due to the adjustment limitation of Goniometer stage, especially stage, the initial desired motion region cannot be measured.
The modified desired motion region is set as qm={[ ]T: -16 ≤ ≤ 16 and -19 ≤ ≤ 19 degree }.
Figure 2.35 shows experiment result using ANN. Figure 2.35 (a) is the result of the all 81data in the orientation domain, where blue and red circle are the reference by Goniometer stage and the result of ANN respectively. The measurement error between reference and result by ANN is shown in Figure 2.35 (b). Numerical maximum and minimum error of and are summarized in Table 2.12.
The predicted measurement accuracy is same as the result of closed circles with rc=14 in Table 2.6 since four closed circles with rc=14 are connected. The measurement accuracy of exmaple 2 using four sensors is within the range of closed circles with rc=14.
(a) Result of all 81data in the orientation domain
(b) Measurement error of 81 data
Figure 2.35 Experiment measurement error of example 2
Table 2.12 Numerical error of experiment in example 2 (unit: degree)
qm=[ ]T
mean max mean max
-16 ≤ ≤ 16 -19 ≤ ≤ 19
0.1653 0.4917 0.1347 0.3334