Calculation and correction of magnetic object positioning error caused by magnet

WANG Sansheng,ZHANG Mingji,ZHANG Ning,and GUO Qiang

1.School of Physics,Beihang University,Beijing 100191,China;

2.School of Instrument Science and Opto-Electronics Engineering,Beihang University,Beijing 100191,China

1.Introduction

Magnetic dipole is defined as a pair of poles or a closed loop of electric current with constant magnetic moment when their dimensions are close to zero[1].Accordingly,a magnetic object can be modeled as a dipole at a remote place where the distance is more than 10 times of the object’s dimension.Such magnetic objects may include magnetic particle attached tumor[2],magnetized unexploded ordnance[3–6],steel made vehicle[7]and cracks[8].Thus the problems of detecting,classifying and positioning a magnetic dipole have attracted much attention owing to their wide applications in medical diagnose and treatment,social security,intelligence transportation and nondestructive testing.Starting from 1940s,magnetic anomaly detecting(MAD)is a dominant positioning technique of magnetic objects by analyzing the anomaly signals from magnetic sensor arrays or aircraft routes[9].However,its detecting distance is strongly limited by the earth field fluctuations,electromagnetic noises and even magnetic fields from carrier.Such problems are greatly solved by the appearance of the magnetic object positioning technique based on magnetic field gradient tensor,which includes nine elements in a 3rd order matrix.It is a more comprehensive description of magnetic field and field gradient,and it is insensitive to the orientation noise as well as the fluctuations of the regional background noise[10–12].Besides the above advantages of magnetic gradient tensor,the rapid development of magnetic field measurement accuracy also contributes a lot to magnetic object positioning distance and accuracy[10].In recent years,many other factors are studied to increase the positioning distance and accuracy such as hard and soft iron errors,sensor biases and non-orthogonality of the sensor configuration[13–15].However,the positioning accuracy and its correction method are seldom reported.We find that the magnetic field gradient cannot be accurately obtained due to the non-ignorable dimension of sensors.

Derived from the basic definition of magnetic field gradient,Section 2 shows how the sensor dimension affects the gradient measurement accuracy and how much it does.Then,the numerical simulations give the inherent positioning error of magnetic dipole based on linear solution of Nara et al.[16]caused by inaccuracies gradient measurement.In Section 3,a correction algorithm is firstly proposed and verified to be feasible by simulation and experiments in Section 4.Finally,some conclusions are drawn in Section 5.

2.Measurement of gradient tensor and positioning theory

2.1 Magnetic field gradient measurement error caused by sensor dimension

It is difficult to obtain the exact magnetic gradient tensor unless a group of extremely small sensors are applied.When a field H generated by a certain magnetic source is measured at a test point r,the magnetic gradient tensor may be theoretically defined as

where∂is the partial derivative of the magnetic field in the r direction.n=r/r,n is the unit vector along the r direction,r=|r|is the distance from the test point to the target.i,j and k are unit vectors along the x,y and z directions,respectively.In technical application,each element of G is calculated by differencing outputs of two numbered sensors assembled with fixed distance(baseline length δ).Hereby,the estimated tensormay be rewritten in a limiting form as,where 1m refers to x,y,z and suffixes 1,2 refer to the two sensors.

However,due to the difficulty in satisfying limit conditionthere will be an inherent error between the theoretical gradient tensor G,namely differential gradient,and the estimated gradient tensornamely differencing gradient.The inherent error is briefly illustrated by calculating results in Fig.1 that shows the field H(m,r),produced by a magnetic dipole placed at origin with its moment m=1 Am2,parallel to x axis.Then,the magnetic field at position r from a dipole with moment m can be formulized as

To make the problem simpler,we suppose that the direction of magnetic moment coincides with the x axis.Take∂xHxfor example,supposing that two sensors are assembled at x=1 m and x=3 m to measure the magnetic field gradient at x=2 m,the differential and differencing value of∂xHxmay be represented by the slope of tangent dashed line lAB,kABand that of dotted line lCD,kCD,respectively shown in Fig.1.The line lABis the tangent to the field intensity curve at a range of 2 m,kABis the value of the differential gradient at this point.lCDis a line that intersects the field intensity curve at ranges 1 m and 3 m,kCDis the calculated difference gradient at range 2 m,kAB=0.026 8 and kCD=0.076 9.

Results show an obvious relative inherent error,(kCD−kAB)/kAB×100%,up to 187%.Although this kind of inherent error may be somewhat alleviated by making δ smaller,sensor dimension effects still reduce the measurement accuracy of the magnetic gradient,leading to signi ficant error of magnetic objects positioning based on magnetic gradient tensor.

Fig.1 Reasons of magnetic field gradient measurement

2.2 Magnetic object positioning based on gradient tensor

In 2006,Nara et al.[14]proposed a simple linear equation system(3)with a unique solution to the magnetic dipole’s position by measuring three components of the field and the magnetic field gradient tensor simultaneously.

Once the position is determined,the moment m can be derived as

where T is the transform matrix defined as

Equation(3)indicates an accurate position r(real distance)of the magnetic dipole as long as the gradient G and the field H are precisely measured.However,the determined position r∗may vary around r because of the inherent error of G−G mentioned in Section 2.Thus,it will also lead to an offset on derived moment m∗to m according to(4).

Fig.2 Systematic error and its corrected results

3.Positioning error correction

To start the correction,the 1st step is substituting the obtained r∗and m∗into(1)to get a virtual field H1and field gradient G1.Then,at the 2nd step,a new position r1and moment m1may be derived by using H1,G1with(4)and(5).After that,the 3rd step is repeating the 1st and 2nd steps to get r2and m2on bias of H2,G2.Thus,in the 4th step,we may estimate the possible e1=r2−r1around r∗.Finally,by adjusting e1to r∗,a more accurate positionis obtained.Repeatsteps 1–5 until the minus of corrected positions between two cycles is small enough.Fortunately,the correction algorithm is simultaneously verified to be convergent in the typical direction as is shown in thick lines in Fig.2.Comparing the thin lines and thick lines in Fig.2,it is obvious to see that the corrected positioning error is less than 100 dB.To make it more comprehensive,i+j+k direction when r=5δ in Fig.2 is chosen as an example and the iteration results of each step are listed in Table 1.

Table 1 Iteration results of each step

Fig.3 Comparison between corrected and uncorrected errors as a function of relative detecting distance considering the measurement noise

Fig.4 Comparison between corrected and uncorrected errors as a function of dipole angle considering the measurement noise

4.Experiment

A coil(average area S=0.2 m,n=320 turns)is set in a 1.2 m×1.2 m platform as a magnetic target,which is shown in Fig.5.The coil is excited by the operational power amplifier(KEPCO-BOP 50–8ML),whose inputsignal is provided by the reference channel from LIA(lockedin amplifier,signal recover-7265).A sampling resistance,rs=0.1 Ohm,is a series connected between the coil and the operational power amplifier to monitor the coil current.Adjust the Kepco power amplifier to achieve the current is 222.4 mA@1 kHz.Thus,the magnetic moment strength of the coil is calculated as 2.44 Am2@1 kHz according to the equation m=nImS.The full tensor magnetic gradiometer is configured by nine pairs of counter wounded copper coils on printcircuit board with half baseline length δ=0.1 m.The induced voltage output of each gradiometer is measured by the input channels of LIA separately.To simplify the experiment,the target coil is moved along the x axis of the platform from x=0.3 m to x=0.9 m keeping its moment m parallel to the i direction.

Fig.5 The experiment platform

Fig.6 Comparison between the experimental uncorrected and corrected positioning error as a function of R

Fig.7 Comparison between the experimental uncorrected and corrected positioning error as a function of θ

5.Conclusions

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