- Email: [email protected]
Integrated calibration and magnetic disturbance compensation of three-axis magnetometers
Accepted Manuscript Integrated calibration and magnetic disturbance compensation of three-axis magnetometers Hongfeng Pang, Jinfei Chen, Ji Li, Qi Zhang, Shitu Luo, Mengchun Pan PII: DOI: Reference:
S0263-2241(16)30348-7 http://dx.doi.org/10.1016/j.measurement.2016.06.056 MEASUR 4176
To appear in:
Measurement
Received Date: Accepted Date:
25 September 2013 24 June 2016
Please cite this article as: H. Pang, J. Chen, J. Li, Q. Zhang, S. Luo, M. Pan, Integrated calibration and magnetic disturbance compensation of three-axis magnetometers, Measurement (2016), doi: http://dx.doi.org/10.1016/ j.measurement.2016.06.056
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Integrated calibration and magnetic disturbance compensation of three-axis magnetometers Hongfeng Pang, Jinfei Chen, Ji Li, Qi Zhang, Shitu Luo and Mengchun Pan College of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha 410073, People’s Republic of China E-mail: [email protected] Abstract: Technological limitations in sensor manufacturing and unwanted magnetic fields will corrupt the measurements of three-axis magnetometers. An experiment with four different magnetic disturbance situations is designed, and the influence of hard-iron and soft-iron material is analyzed. The calibration method with magnetic disturbance parameters is proposed for calibration and magnetic disturbance compensation of three axis magnetometers. It is not necessary to compute pseudo-linear parameters, thus the integrated parameters are computed directly by solving nonlinear equations. To employ this method, a nonmagnetic rotation equipment, a CZM-3 proton magnetometer, a DM-050 three-axis magnetometer, two magnets and two steel tubes are used. Calibration performance is discussed in the four situations. Compared with several traditional calibration methods, experiment results show that the proposed method has better integrated compensation performance in all situations, and error is reduced by several orders of magnitude. After compensation, RMS error is reduced from 10797.962 nT to 15.309 nT when the big magnet and steel tube are deployed. It suggests an useful method for calibration and magnetic disturbance compensation of three-axis magnetometers. Keywords: three-axis magnetometer, calibration, magnetic disturbance compensation, hard-iron, soft-iron, nonlinear equations.
1．Introduction Three-axis magnetometers are widely used in satellites, aircraft navigation, autonomous underwater vehicles (AUV) navigation [1, 2]. The accuracy of three-axis magnetometers is limited by different scale, bias of each axis and
1
non-orthogonality between axes [3, 4]. In addition, in most practical applications there will be unwanted magnetic fields corrupting the measurements of the magnetometer. Therefore, it is important to calibrate magnetometer and compensate unwanted magnetic fields. Risbo et al. introduced the ‘thin shell’ calibration procedure and developed spherical harmonic modeling. They used a test coil system and an overhauser absolute scalar proton magnetometer. Parameters are determined by measuring the current in the coils [5]. Petrucha et al. constructed a nonmagnetic calibration platform to calibrate magnetometers [6]. Renk et al. used a robotic system arm to calibrate magnetometers [7]. Zhu et al. used a specified reference rectangular coordinate frame to calibrate magnetometers, with the help of an optical system [8]. Pang et al. proposed differential evolution algorithm for the calibration of three-axis magnetometers [9]. Merayo et al. introduced a linear least squares estimator to find the parameters independently and uniquely for a given data [10]. Wang et al. used neural networks to calibrate magnetometers in compass with an external heading reference [11]. Constrained Newton optimization method was also used to solve this non-linear minimization problem directly in the calibration parameter space [12, 13]. Jurman et al. used Quasi-Newton (QN) algorithm. In order to compute calibration parameters, the initial values of the parameters and the constraints need to be set according to the typical values quoted in the sensors’ datasheets [14]. The calibration method by EKF was proved to be applicable in limited cases when the biases and initial heading errors are not large, since the first-order approximation of the EKF can not adequately capture the large linearization errors [15]. Crassidis et al. compared several real-time algorithms for the calibration of three-axis magnetometers performing on-orbit during typical spacecraft mission-mode operations, and unscented Kalman filtering (UKF) algorithm demonstrated better performance [16]. Pang et al. used a nonlinear least squares method to calibrate three axis magnetometers [17]. However, these mentioned calibration methods mainly considered technological limitations in sensor manufacturing, and the calibration model mainly contained interior sensor error: scale factor error, offset and non-orthogonality error. The influence of magnetic disturbance in calibration model was usually ignored.
2
In fact, the environment around magnetometers affects the nature of output errors. The magnetic field distortions occur in the presence of ferromagnetic elements found in the vicinity of the sensor and due to devices mounted in the vehicle’s structure [18]. Constant biases or null shifts caused by magnetically hard materials are sometimes referred to as hard-iron biases, whereas biases due to magnetic induction from magnetically soft materials are called soft-iron errors [19]. Some researchers considered magnetic disturbance parameters in magnetometer calibration model, and ellipsoid fitting methods are proposed for compensation in some applications. Gebre-Egziabher et al. proposed a two-step method to calibrate magnetometer offset, hard-iron bias and scale factor error based on ellipsoid fitting, which ignored non-orthogonality and soft-iron error [20]. Foster et al. presented an extension of the nonlinear two-step estimation algorithm originally developed for the calibration of solid-state strapdown magnetometers [21]. Pylvanainen et al. introduced recursive least squares (RLS) algorithm for ellipsoid fitting [22]. Using the ellipsoid-fitting methods, it is important to obtain the ellipsoid-fitting parameters in the first step, and the measured data must in the Euclidean space [19]. The first step is the classic least-squares problem which introduced a non-uniform weighting of the various measurements. In the error-free case, the traditional least-squares estimator (or its stochastic derivatives such as the Kalman filter) would be inappropriate; they would yield the trivial solution [23]. Total least-squares (TLS), batch least-squares (BLS) or recursive least squares (RLS) are proposed to compute ellipsoid-fitting parameters [19, 22, 23]. Similarly, this would have led to nonequivalent optimization problems though. Thus, the original optimization problem will be distorted. In some cases this can be a problem [24]. In addition, it should be mentioned that linearization parameters may be optimal in a least-squares sense, the true magnetometer parameters may not necessarily be so [21]. As for the analysis of material influence, Gebre-Egziabher compared the influence of different soft-iron material in simulation [23]. But little work has been done to quantitatively analyze the influence of different hard-iron and soft-iron material in experiment, and the comparison of calibration performance with different magnetic disturbance is absent.
3
In this paper, a new calibration method is proposed for calibration and magnetic disturbance compensation of three axis magnetometers. It is with better calibration performance compared with traditional calibration methods. An experiment with four different magnetic disturbance situations is designed, and the influence of hard-iron and soft-iron material is analyzed. Compared with several calibration methods, the proposed method is with better integrated compensation performance in all situations.
2. Calibration and compensation theory A comprehensive mathematical model for the output error of a magnetometer is given as [23]:
0 0 1 1 sx B Cs C C H b 0 1 sy 0 z 0 0 1 sz y
z 1 x
y x 1
1 xx xy xz H x bx x 1 yy yz H y by y yx zx zy 1 zz H z bz z
(1)
Equation (1) can be expressed as: Bx 1 sx B (1 s ) y z y Bz (1 sz ) y
(1 sx ) z 1 sy (1 sz ) x
(1 sx ) y 1 xx xy xz H x bx x (1 s y ) x yx 1 yy yz H y by y 1 sz zx zy 1 zz H z bz z
(2)
Equation (2) can be expressed as: Bx C11 C12 B C y 21 C22 Bz C31 C32
C13 H x bx x C23 H y by y C33 H z bz z
(3)
Then, the integrated compensation model can be expressed as: H x D11 H q y 21 H z q31
q12 D22 q32
q13 Bx bx1 q23 By by1 D33 Bz bz1
(4)
There are 12 unknown parameters in equation (4): bx1 , by1 , bz1 are integrated offset of X axis, Y axis, and Z axis, respectively; D11 , D22 , D33 are integrated scale factor of X axis, Y axis, and Z axis, respectively; the six off-diagonal elements q12 , q21 , q13 , q31 , q23 , q32 are not symmetric because of soft-iron error. Equation (4) can be expressed as: 4
H x =D11 （Bx bx1） +q12 （By by1） +q13 （Bz bz1） （Bx bx1） +D22（By by1） +q23 （Bz bz1） H y =q21 H =q（B b ） （By by1） +D33（Bz bz1） z 31 x x1 +q32
(5)
A nonlinear equation can be obtained by squaring both side of equation (5): H 2 H x2 H y2 H z2 [ D11 （Bx bx1） +q12（By by1） +q13 （Bz bz1） ]2 [q21 （Bx bx1） +D22（By by1） +q23 （Bz bz1） ]2
(6)
[q31 （Bx bx1） +q32（By by1） +D33（Bz bz1） ]2
N nonlinear equations can be established when a batch of data are measured (data number is N). There are 12 parameters: D11 , D22 , D33 , q12 , q13 , q23 , q21 , q31 , q32 , bx1 , by1 , bz1 . So, at least 12 data should be measured. The function “ fsolve” in Matlab is used to solve nonlinear equations, which is widely used to find the root of a system of nonlinear equations. Compared with ellipsoid-fitting method, it is not necessary to compute pseudo-linear parameters, thus it can avoid the problem mentioned in ref. [23, 24].
3. Experiment 3.1 Experimental system The experiment system consists of a three-axis fluxgate magnetometer (DM-050), a CZM-3 proton magnetometer, a two dimensional nonmagnetic rotation equipment, a portable computer, a 12V-DC portable power device, two magnets, two steel tubes, data acquisition and process software. The proton magnetometer is used to provide the value of magnetic field intensity. The portable computer is connected with the fluxgate magnetometer, which is used for data storage. The portable power device is required to provide power for fluxgate magnetometer. The two magnets are used as hard-iron material (small magnet and big magnet). The two steel tubes are used as soft-iron material (small steel tube and big steel tube). The inner and outer diameter of small steel tube (soft-iron 1) is 35 mm and 38 mm respectively, and its length is 120 mm. The inside and outside diameter of big steel tube (soft-iron 2) is 32 mm and 40 mm respectively, and its length is 200 mm. The small magnet (hard-iron 1) is annular; the inside and outside diameter of is 32 mm and 60 mm respectively, and the thick is 8 mm. The big magnet (hard-iron 2) is solid; the diameter is 50 mm, and the thick is 5
10 mm. Hard-iron material and soft-iron material are shown in Fig. 1. Data acquisition software of DM-050 magnetometer is STL GradMag. STL GradMag is the official software given by STL company, which is also used for signal display and data storage. According to the manual of the DM-050 fluxgate magnetometer, its measurement range is ±1000000nT (vector value); offset <5 nT; non-orthogonality angle <0.01°; scale factor error < 0.001. As for CZM-3 proton magnetometer, its measurement range is from 30000 nT to 70000 nT (scalar value); resolution is 0.1 nT; absolute accuracy is ±1 nT.
Fig. 1. Hard-iron and soft-iron material.
3.2 Experimental operation As shown in Fig. 2, the distance between DM-050 magnetometer and soft-iron is 300 mm, and hard-iron is 150 mm from soft-iron. The distance between magnetometer, hard-iron and soft-iron is fixed, but there are four magnetic disturbance situations: (1) hard-iron 1 and soft-iron 1. (2) hard-iron 1 and soft-iron 2. (3) hard-iron 2 and soft-iron 1. (4) hard-iron 2 and soft-iron 2. In order to compare the magnetic disturbance of different hard-iron material and soft-iron material, the 36 static measurement strategy in 3D is implemented to test magnetic disturbance, and DM-050 magnetometer poses are identical in the four situations by controlling the nonmagnetic rotation equipment. The details can be described as
6
follows [17]: (1) The rotation table is kept to be horizontal by adjusting its base shelves, both its horizontal rotation angle and upright rotation angle are adjusted to be zero. (2) The magnetic field is measured and saved in computer by acquisition software about one minute. The average is considered as measured value. (3) The horizontal rotation angle is fixed and the upright rotation angle is increased by 60°. (4) The horizontal rotation angle is increased by 60°when the upright rotation angle became 360°( 0°). The procedure (2) to (4) are repeated until 36 measured vectors are obtained at the calibration position.
Fig. 2. Experimental system.
3.3 Experimental results Fig. 3, 4, 5 and 6 show the calibration performance in situation 1, 2, 3 and 4 respectively. It is clear that the integrated compensation performance is good, and error is reduced by several orders after calibration. The details of magnetic field disturbance in the four situations are shown in table 1. It shows that original disturbance is mainly influenced by hard-iron, which is more than ten thousands nT in situation 3 and 4 when the big magnet is deployed. Such big magnetic field disturbance is intolerant. It should be mentioned that the precision of nonmagnetic rotation equipment setting is not critical, so magnetometer poses are not exactly coincident in the four situations, and that is why 7
the original disturbance of situation 4 is slightly less than situation 3. However, hard-iron material is not the key factor influencing integrated compensation result. Comparing the calibration performance of situation 1 with situation 3, it is clear that the calibration performance is similar in different hard-iron disturbance situations. After calibration, RMS error is reduced from 1632.317 nT and 10797.962 nT to 15.626 nT and 15.309 nT respectively. As shown in situation 2 and situation 4, RMS error after calibration is about one hundred nT when the big soft-iron tube is used. So, the calibration performance is mainly limited by soft-iron material. On the whole, the proposed method has good performance about integrated compensation in the four magnetic disturbance situations. In addition, magnetic disturbance is the major contribution of integrated parameters which can be directly computed by nonlinear equations, so it is convenient and effective to analyze separately the influence of soft-iron and hard-iron material.
Fig. 3. Integrated compensation performance in situation 1 (hard-iron 1 and soft-iron 1).
8
Fig. 4. Integrated compensation performance in situation 2 (hard-iron 1 and soft-iron 2).
Fig. 5. Integrated compensation performance in situation 3 (hard-iron 2 and soft-iron 1).
9
Fig. 6. Integrated compensation performance in situation 4 (hard-iron 2 and soft-iron 2).
Table 1. Compensation results in four situations with different methods. RMS error
Situation 1 (nT)
Situation 2 (nT)
Situation 3 (nT)
Situation 4 (nT)
Before compensation
1632.317
3366.264
10797.962
10533.936
The proposed method
15.626
79.370
15.309
100.998
RLS
24.857
96.3607
30.774
143.565
UKF
35.699
122.929
14562.060
31929.542
In addition, the compensation results of different methods are shown in Table 1. The details of RLS for ellipsoid-fitting and UKF are illustrated in [22] and [16] respectively. It shows that the proposed method has better compensation performance in all situations, and UKF is distorted in situation 3 and 4 when the big hard-iron is deployed. There are two reasons: (1) The compensation performance of RLS and UKF algorithms is limited by initial parameters which should be carefully chosen. (2) The robustness of the proposed method is better.
4. Conclusions The accuracy of three-axis magnetometers is limited by different scale, bias of each axis and non-orthogonality between axes. In addition, the magnetic field reading distortions occur in the presence of ferromagnetic elements found in the vicinity of three magnetometers. The influence of hard-iron and soft-iron material is analyzed in experiment. It
10
shows that original disturbance is mainly influenced by hard-iron, which is more than ten thousands nT when a big magnet is deployed. However, hard-iron material is not the key factor influencing compensation result, and the compensation performance is similar in different hard-iron disturbance situations. So, the compensation performance is mainly limited by soft-iron material. Compared with RLS and UKF, the proposed method has better performance about integrated compensation in all magnetic disturbance situations.
Acknowledgments This work is sponsored by National Natural Science Foundation of China (project number: 51175507). The authors would like to extend their appreciation to college of Mechatronics Engineering and Automation, National University of Defense Technology for their contributions to the work.
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Highlights
1. 2. 3. 4. 5.
The influence of hard-iron and soft-iron material is analyzed. The proposed method has good compensation performance in all situations. Compared with ellipsoid-fitting method, it is not necessary to compute pseudo-linear parameters. Compared with several traditional methods, the proposed method has better integrated compensation performance. The method can be used for integrated compensation in underwater vehicles or aircrafts.
13
S0263-2241(16)30348-7 http://dx.doi.org/10.1016/j.measurement.2016.06.056 MEASUR 4176
To appear in:
Measurement
Received Date: Accepted Date:
25 September 2013 24 June 2016
Please cite this article as: H. Pang, J. Chen, J. Li, Q. Zhang, S. Luo, M. Pan, Integrated calibration and magnetic disturbance compensation of three-axis magnetometers, Measurement (2016), doi: http://dx.doi.org/10.1016/ j.measurement.2016.06.056
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Integrated calibration and magnetic disturbance compensation of three-axis magnetometers Hongfeng Pang, Jinfei Chen, Ji Li, Qi Zhang, Shitu Luo and Mengchun Pan College of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha 410073, People’s Republic of China E-mail: [email protected] Abstract: Technological limitations in sensor manufacturing and unwanted magnetic fields will corrupt the measurements of three-axis magnetometers. An experiment with four different magnetic disturbance situations is designed, and the influence of hard-iron and soft-iron material is analyzed. The calibration method with magnetic disturbance parameters is proposed for calibration and magnetic disturbance compensation of three axis magnetometers. It is not necessary to compute pseudo-linear parameters, thus the integrated parameters are computed directly by solving nonlinear equations. To employ this method, a nonmagnetic rotation equipment, a CZM-3 proton magnetometer, a DM-050 three-axis magnetometer, two magnets and two steel tubes are used. Calibration performance is discussed in the four situations. Compared with several traditional calibration methods, experiment results show that the proposed method has better integrated compensation performance in all situations, and error is reduced by several orders of magnitude. After compensation, RMS error is reduced from 10797.962 nT to 15.309 nT when the big magnet and steel tube are deployed. It suggests an useful method for calibration and magnetic disturbance compensation of three-axis magnetometers. Keywords: three-axis magnetometer, calibration, magnetic disturbance compensation, hard-iron, soft-iron, nonlinear equations.
1．Introduction Three-axis magnetometers are widely used in satellites, aircraft navigation, autonomous underwater vehicles (AUV) navigation [1, 2]. The accuracy of three-axis magnetometers is limited by different scale, bias of each axis and
1
non-orthogonality between axes [3, 4]. In addition, in most practical applications there will be unwanted magnetic fields corrupting the measurements of the magnetometer. Therefore, it is important to calibrate magnetometer and compensate unwanted magnetic fields. Risbo et al. introduced the ‘thin shell’ calibration procedure and developed spherical harmonic modeling. They used a test coil system and an overhauser absolute scalar proton magnetometer. Parameters are determined by measuring the current in the coils [5]. Petrucha et al. constructed a nonmagnetic calibration platform to calibrate magnetometers [6]. Renk et al. used a robotic system arm to calibrate magnetometers [7]. Zhu et al. used a specified reference rectangular coordinate frame to calibrate magnetometers, with the help of an optical system [8]. Pang et al. proposed differential evolution algorithm for the calibration of three-axis magnetometers [9]. Merayo et al. introduced a linear least squares estimator to find the parameters independently and uniquely for a given data [10]. Wang et al. used neural networks to calibrate magnetometers in compass with an external heading reference [11]. Constrained Newton optimization method was also used to solve this non-linear minimization problem directly in the calibration parameter space [12, 13]. Jurman et al. used Quasi-Newton (QN) algorithm. In order to compute calibration parameters, the initial values of the parameters and the constraints need to be set according to the typical values quoted in the sensors’ datasheets [14]. The calibration method by EKF was proved to be applicable in limited cases when the biases and initial heading errors are not large, since the first-order approximation of the EKF can not adequately capture the large linearization errors [15]. Crassidis et al. compared several real-time algorithms for the calibration of three-axis magnetometers performing on-orbit during typical spacecraft mission-mode operations, and unscented Kalman filtering (UKF) algorithm demonstrated better performance [16]. Pang et al. used a nonlinear least squares method to calibrate three axis magnetometers [17]. However, these mentioned calibration methods mainly considered technological limitations in sensor manufacturing, and the calibration model mainly contained interior sensor error: scale factor error, offset and non-orthogonality error. The influence of magnetic disturbance in calibration model was usually ignored.
2
In fact, the environment around magnetometers affects the nature of output errors. The magnetic field distortions occur in the presence of ferromagnetic elements found in the vicinity of the sensor and due to devices mounted in the vehicle’s structure [18]. Constant biases or null shifts caused by magnetically hard materials are sometimes referred to as hard-iron biases, whereas biases due to magnetic induction from magnetically soft materials are called soft-iron errors [19]. Some researchers considered magnetic disturbance parameters in magnetometer calibration model, and ellipsoid fitting methods are proposed for compensation in some applications. Gebre-Egziabher et al. proposed a two-step method to calibrate magnetometer offset, hard-iron bias and scale factor error based on ellipsoid fitting, which ignored non-orthogonality and soft-iron error [20]. Foster et al. presented an extension of the nonlinear two-step estimation algorithm originally developed for the calibration of solid-state strapdown magnetometers [21]. Pylvanainen et al. introduced recursive least squares (RLS) algorithm for ellipsoid fitting [22]. Using the ellipsoid-fitting methods, it is important to obtain the ellipsoid-fitting parameters in the first step, and the measured data must in the Euclidean space [19]. The first step is the classic least-squares problem which introduced a non-uniform weighting of the various measurements. In the error-free case, the traditional least-squares estimator (or its stochastic derivatives such as the Kalman filter) would be inappropriate; they would yield the trivial solution [23]. Total least-squares (TLS), batch least-squares (BLS) or recursive least squares (RLS) are proposed to compute ellipsoid-fitting parameters [19, 22, 23]. Similarly, this would have led to nonequivalent optimization problems though. Thus, the original optimization problem will be distorted. In some cases this can be a problem [24]. In addition, it should be mentioned that linearization parameters may be optimal in a least-squares sense, the true magnetometer parameters may not necessarily be so [21]. As for the analysis of material influence, Gebre-Egziabher compared the influence of different soft-iron material in simulation [23]. But little work has been done to quantitatively analyze the influence of different hard-iron and soft-iron material in experiment, and the comparison of calibration performance with different magnetic disturbance is absent.
3
In this paper, a new calibration method is proposed for calibration and magnetic disturbance compensation of three axis magnetometers. It is with better calibration performance compared with traditional calibration methods. An experiment with four different magnetic disturbance situations is designed, and the influence of hard-iron and soft-iron material is analyzed. Compared with several calibration methods, the proposed method is with better integrated compensation performance in all situations.
2. Calibration and compensation theory A comprehensive mathematical model for the output error of a magnetometer is given as [23]:
0 0 1 1 sx B Cs C C H b 0 1 sy 0 z 0 0 1 sz y
z 1 x
y x 1
1 xx xy xz H x bx x 1 yy yz H y by y yx zx zy 1 zz H z bz z
(1)
Equation (1) can be expressed as: Bx 1 sx B (1 s ) y z y Bz (1 sz ) y
(1 sx ) z 1 sy (1 sz ) x
(1 sx ) y 1 xx xy xz H x bx x (1 s y ) x yx 1 yy yz H y by y 1 sz zx zy 1 zz H z bz z
(2)
Equation (2) can be expressed as: Bx C11 C12 B C y 21 C22 Bz C31 C32
C13 H x bx x C23 H y by y C33 H z bz z
(3)
Then, the integrated compensation model can be expressed as: H x D11 H q y 21 H z q31
q12 D22 q32
q13 Bx bx1 q23 By by1 D33 Bz bz1
(4)
There are 12 unknown parameters in equation (4): bx1 , by1 , bz1 are integrated offset of X axis, Y axis, and Z axis, respectively; D11 , D22 , D33 are integrated scale factor of X axis, Y axis, and Z axis, respectively; the six off-diagonal elements q12 , q21 , q13 , q31 , q23 , q32 are not symmetric because of soft-iron error. Equation (4) can be expressed as: 4
H x =D11 （Bx bx1） +q12 （By by1） +q13 （Bz bz1） （Bx bx1） +D22（By by1） +q23 （Bz bz1） H y =q21 H =q（B b ） （By by1） +D33（Bz bz1） z 31 x x1 +q32
(5)
A nonlinear equation can be obtained by squaring both side of equation (5): H 2 H x2 H y2 H z2 [ D11 （Bx bx1） +q12（By by1） +q13 （Bz bz1） ]2 [q21 （Bx bx1） +D22（By by1） +q23 （Bz bz1） ]2
(6)
[q31 （Bx bx1） +q32（By by1） +D33（Bz bz1） ]2
N nonlinear equations can be established when a batch of data are measured (data number is N). There are 12 parameters: D11 , D22 , D33 , q12 , q13 , q23 , q21 , q31 , q32 , bx1 , by1 , bz1 . So, at least 12 data should be measured. The function “ fsolve” in Matlab is used to solve nonlinear equations, which is widely used to find the root of a system of nonlinear equations. Compared with ellipsoid-fitting method, it is not necessary to compute pseudo-linear parameters, thus it can avoid the problem mentioned in ref. [23, 24].
3. Experiment 3.1 Experimental system The experiment system consists of a three-axis fluxgate magnetometer (DM-050), a CZM-3 proton magnetometer, a two dimensional nonmagnetic rotation equipment, a portable computer, a 12V-DC portable power device, two magnets, two steel tubes, data acquisition and process software. The proton magnetometer is used to provide the value of magnetic field intensity. The portable computer is connected with the fluxgate magnetometer, which is used for data storage. The portable power device is required to provide power for fluxgate magnetometer. The two magnets are used as hard-iron material (small magnet and big magnet). The two steel tubes are used as soft-iron material (small steel tube and big steel tube). The inner and outer diameter of small steel tube (soft-iron 1) is 35 mm and 38 mm respectively, and its length is 120 mm. The inside and outside diameter of big steel tube (soft-iron 2) is 32 mm and 40 mm respectively, and its length is 200 mm. The small magnet (hard-iron 1) is annular; the inside and outside diameter of is 32 mm and 60 mm respectively, and the thick is 8 mm. The big magnet (hard-iron 2) is solid; the diameter is 50 mm, and the thick is 5
10 mm. Hard-iron material and soft-iron material are shown in Fig. 1. Data acquisition software of DM-050 magnetometer is STL GradMag. STL GradMag is the official software given by STL company, which is also used for signal display and data storage. According to the manual of the DM-050 fluxgate magnetometer, its measurement range is ±1000000nT (vector value); offset <5 nT; non-orthogonality angle <0.01°; scale factor error < 0.001. As for CZM-3 proton magnetometer, its measurement range is from 30000 nT to 70000 nT (scalar value); resolution is 0.1 nT; absolute accuracy is ±1 nT.
Fig. 1. Hard-iron and soft-iron material.
3.2 Experimental operation As shown in Fig. 2, the distance between DM-050 magnetometer and soft-iron is 300 mm, and hard-iron is 150 mm from soft-iron. The distance between magnetometer, hard-iron and soft-iron is fixed, but there are four magnetic disturbance situations: (1) hard-iron 1 and soft-iron 1. (2) hard-iron 1 and soft-iron 2. (3) hard-iron 2 and soft-iron 1. (4) hard-iron 2 and soft-iron 2. In order to compare the magnetic disturbance of different hard-iron material and soft-iron material, the 36 static measurement strategy in 3D is implemented to test magnetic disturbance, and DM-050 magnetometer poses are identical in the four situations by controlling the nonmagnetic rotation equipment. The details can be described as
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follows [17]: (1) The rotation table is kept to be horizontal by adjusting its base shelves, both its horizontal rotation angle and upright rotation angle are adjusted to be zero. (2) The magnetic field is measured and saved in computer by acquisition software about one minute. The average is considered as measured value. (3) The horizontal rotation angle is fixed and the upright rotation angle is increased by 60°. (4) The horizontal rotation angle is increased by 60°when the upright rotation angle became 360°( 0°). The procedure (2) to (4) are repeated until 36 measured vectors are obtained at the calibration position.
Fig. 2. Experimental system.
3.3 Experimental results Fig. 3, 4, 5 and 6 show the calibration performance in situation 1, 2, 3 and 4 respectively. It is clear that the integrated compensation performance is good, and error is reduced by several orders after calibration. The details of magnetic field disturbance in the four situations are shown in table 1. It shows that original disturbance is mainly influenced by hard-iron, which is more than ten thousands nT in situation 3 and 4 when the big magnet is deployed. Such big magnetic field disturbance is intolerant. It should be mentioned that the precision of nonmagnetic rotation equipment setting is not critical, so magnetometer poses are not exactly coincident in the four situations, and that is why 7
the original disturbance of situation 4 is slightly less than situation 3. However, hard-iron material is not the key factor influencing integrated compensation result. Comparing the calibration performance of situation 1 with situation 3, it is clear that the calibration performance is similar in different hard-iron disturbance situations. After calibration, RMS error is reduced from 1632.317 nT and 10797.962 nT to 15.626 nT and 15.309 nT respectively. As shown in situation 2 and situation 4, RMS error after calibration is about one hundred nT when the big soft-iron tube is used. So, the calibration performance is mainly limited by soft-iron material. On the whole, the proposed method has good performance about integrated compensation in the four magnetic disturbance situations. In addition, magnetic disturbance is the major contribution of integrated parameters which can be directly computed by nonlinear equations, so it is convenient and effective to analyze separately the influence of soft-iron and hard-iron material.
Fig. 3. Integrated compensation performance in situation 1 (hard-iron 1 and soft-iron 1).
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Fig. 4. Integrated compensation performance in situation 2 (hard-iron 1 and soft-iron 2).
Fig. 5. Integrated compensation performance in situation 3 (hard-iron 2 and soft-iron 1).
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Fig. 6. Integrated compensation performance in situation 4 (hard-iron 2 and soft-iron 2).
Table 1. Compensation results in four situations with different methods. RMS error
Situation 1 (nT)
Situation 2 (nT)
Situation 3 (nT)
Situation 4 (nT)
Before compensation
1632.317
3366.264
10797.962
10533.936
The proposed method
15.626
79.370
15.309
100.998
RLS
24.857
96.3607
30.774
143.565
UKF
35.699
122.929
14562.060
31929.542
In addition, the compensation results of different methods are shown in Table 1. The details of RLS for ellipsoid-fitting and UKF are illustrated in [22] and [16] respectively. It shows that the proposed method has better compensation performance in all situations, and UKF is distorted in situation 3 and 4 when the big hard-iron is deployed. There are two reasons: (1) The compensation performance of RLS and UKF algorithms is limited by initial parameters which should be carefully chosen. (2) The robustness of the proposed method is better.
4. Conclusions The accuracy of three-axis magnetometers is limited by different scale, bias of each axis and non-orthogonality between axes. In addition, the magnetic field reading distortions occur in the presence of ferromagnetic elements found in the vicinity of three magnetometers. The influence of hard-iron and soft-iron material is analyzed in experiment. It
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shows that original disturbance is mainly influenced by hard-iron, which is more than ten thousands nT when a big magnet is deployed. However, hard-iron material is not the key factor influencing compensation result, and the compensation performance is similar in different hard-iron disturbance situations. So, the compensation performance is mainly limited by soft-iron material. Compared with RLS and UKF, the proposed method has better performance about integrated compensation in all magnetic disturbance situations.
Acknowledgments This work is sponsored by National Natural Science Foundation of China (project number: 51175507). The authors would like to extend their appreciation to college of Mechatronics Engineering and Automation, National University of Defense Technology for their contributions to the work.
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Highlights
1. 2. 3. 4. 5.
The influence of hard-iron and soft-iron material is analyzed. The proposed method has good compensation performance in all situations. Compared with ellipsoid-fitting method, it is not necessary to compute pseudo-linear parameters. Compared with several traditional methods, the proposed method has better integrated compensation performance. The method can be used for integrated compensation in underwater vehicles or aircrafts.
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