Three-dimensional localization of a rotating magnetic dipole from the Fourier integrals of its magnetic flux density with acceleration data

Indoor localization of objects when these bodies are affected by occlusion remains challenging. This paper presents a novel method to estimate the position of a magnetic marker that can be attached to objects such as a robot or a moving vehicle for indoor localization. We derive a formula to compute the azimuth and zenith angles of and the distance to a rotating magnet from the Fourier components of the magnetic flux density at the marker’s rotational frequency. The proposed method does not require temporal changes in the direction of the magnetic moment. As a result, the method does not require communication between the sensor and the marker. Also, using a Fourier component enhances the robustness of the method to environmental noise. The method is verified experimentally and shows an average error of 49.9 mm for estimation of the three-dimensional position of the marker in a 4000 mm by 2000 mm domain where z = 350 mm. In addition, we verified that the marker position can be estimated even if the sensor is tilted with respect to the horizontal plane using the sensor’s acceleration data.Indoor localization of objects when these bodies are affected by occlusion remains challenging. This paper presents a novel method to estimate the position of a magnetic marker that can be attached to objects such as a robot or a moving vehicle for indoor localization. We derive a formula to compute the azimuth and zenith angles of and the distance to a rotating magnet from the Fourier components of the magnetic flux density at the marker’s rotational frequency. The proposed method does not require temporal changes in the direction of the magnetic moment. As a result, the method does not require communication between the sensor and the marker. Also, using a Fourier component enhances the robustness of the method to environmental noise. The method is verified experimentally and shows an average error of 49.9 mm for estimation of the three-dimensional position of the marker in a 4000 mm by 2000 mm domain where z = 350 mm. In addition, we verified that the marker position can be estimated even if the sensor ...


I. INTRODUCTION
Indoor localization is important for various applications such as navigation. 1 However, the fact that the global positioning system couldn't be used indoors presented a problem. 2 Various methods have been proposed to estimate the indoor position using RFID tags, 3 geomagnetism, 4 and so on. While methods based on simultaneous localization and mapping were proposed, 5-7 they required map reconstruction when the environment is changed. Also, electromagnetic-wave-based systems have been proposed by Koo 8 and Feng 9 in which the received signal strength of electromagnetic waves was measured. However, the electromagnetic-wave-based method is limited in terms of accuracy because of multi-path effects. 2 When considering the localization of vehicles moving inside a factory or warehouse for preventing human contact accidents, in which there are both people and metal shelves, localization with an accuracy of several tens of millimeters within a range of several square meters would be required.
When compared with these methods, a low-frequency, magnetic-field-based method offers the advantage that it is less strongly affected by both nonferromagnetic objects and human bodies. The conventional method used for localization of a magnetic dipole involves the use of an array of magnetic sensors and dipole localization using the nonlinear least squares fitting method. [10][11][12] In addition, methods using a magnetic marker that is rotated electrically 13 or mechanically 14 have been proposed. Paperno et al. 13 generated a rotating magnetic field by inducing quadrature signals in two orthogonal coils. They then estimated the three-dimensional position of a pickup coil by detecting the times at which the norm of the magnetic field vector reaches a maximum and a minimum at the sensor position. Watanabe et al. 14 proposed a method in which a spinning magnet and a smartphone are used as a magnetic marker and a three-axis magnetic field sensor, respectively. The sensor's azimuth angle is determined by the direction of the rotating magnet when the measured magnetic field strength is at its maximum. The elevation angle and the distance between the magnet and the smartphone are also computed from the magnetic field components. However,

ARTICLE
scitation.org/journal/adv in all these methods, the temporal changes in the direction of the magnetic moment, which is rotated either electrically or mechanically, must be detected. In other words, the phase of the rotating magnetic moment must be known to determine the position of the sensor relative to the marker. This requires communication between the marker and the sensor. Additionally, detection of the maximum magnetic field strength is strongly affected by the signal-to-noise ratio.
In this study, we propose a method to estimate the position of the rotating marker relative to the sensor without knowledge of the temporal changes in the direction of the magnetic moment, which thus eliminates the necessity for communication between sensor and marker. In our method, the azimuth and zenith angles and the distance to the marker are expressed explicitly in terms of the Fourier components of the magnetic flux density at the rotating marker's frequency. Therefore, by quadrature detection of the measured magnetic flux density, the marker position can be estimated without acquisition of the phase of the rotating magnetic moment. In our previous work, 15 the azimuth angle was restricted to the range from 0 to π; this restriction allowed the sensor to be set on a wall in a room. In this paper, we propose a method that allows this restriction to be removed.

A. Problem setting
A sensor unit composed of three-axis magnetic sensors and three-axis accelerometers is placed at the origin. The global coordinate system is defined as follows: the z-axis is oriented in the vertically upward direction using three-axis accelerometers and the x-axis is oriented toward the north using the three-axis magnetic sensors. As shown in Fig. 1, a magnetic marker exists at and has a magnetic moment, rotating in the xy-plane, given by where ω is the angular frequency, p is the strength of the magnetic moment, and θ 0 is the initial phase of this moment. In this paper, we assume that 0 < ψ < π 2 . While this restricts the positioning of a marker relative to the sensor, i.e., the marker must exist in the upper space of the sensor, it would still be useful in many situations in practical applications.
Under these conditions, the magnetic flux density measured at the origin is expressed as: where μ 0 is the free space permeability. We aim to identify the azimuth angle ϕ, the zenith angle ψ, and the distance r to the marker when viewed from the sensor using the magnetic flux density given in Eq. (3). Importantly, we do not know the initial phase, and thus the phase, ωt + θ 0 , at time t. ω and p are assumed to be known. In practical situations, ω can be estimated by fast Fourier transform (FFT) using the measured time series of the magnetic flux density. First, for simplicity, we explain our method for the case where the sensor coordinate system (XYZ-axes) coincides with the global coordinate system (xyz-axes). The method that is applicable when these systems are different is presented in section IIC.

B. Proposed method
We derive a formula to estimate ϕ, ψ, and r in terms of the Fourier components of the magnetic flux density at the rotational frequency ω. Let us consider Bx + iBy. From Eq. (3), we obtain (4) Substituting Eq. (1) and (2) to Eq. (4), we find (5) Therefore, by quadrature detection of Bx + iBy, we obtain the Fourier components of the angular frequency ± ω as and where T = 2π ω and N is an integer. Also, Bz is expressed as and thus we have From Eqs. (7) and (9), we find ARTICLE scitation.org/journal/adv which are used to determine the zenith and azimuth angles, respectively. The distance between the marker and the sensor can be estimated as follows. From Eqs. (6) and (7), we have Because Eq. (12) is not equal to zero for an arbitrary value of ψ, the following holds.
Therefore, if the strength of the moment, p, is known a priori, we can then estimate the distance r by inserting the measured ψ value into Eq. (13). These equations hold irrespective of the value of θ 0 , illustrating that the position of the magnetic marker relative to the sensor can be determined from Eqs. (10), (11), and (13) using the Fourier components of Bx + iBy and Bz obtained by quadrature detection of the measured magnetic flux density without knowing the initial phase, and thus the phase at any arbitrary time, of the rotating magnetic moment of the marker.

C. Coordinate transformation
In this section, we consider the case where the sensor coordinate system does not coincide with the global coordinate system. Let α, β and γ be the angles that the XYZ-axes make with the xyz-axes, respectively. 16 Let g = g 0 (gx, gy, gz) T be the measured acceleration in the XYZ-coordinate system, where √ g 2 x + g 2 y + g 2 z = 1, and g 0 is the gravitational acceleration. Let B = B 0 (BX, BY , B Z ) T be the measured geomagnetic flux density, where , and B 0 is its strength. Because the gravitational acceleration vector in the xyz-axes is (0, 0, −g 0 ) T , α and β are calculated from (gx, gy, gz) using Also, because the geomagnetism vector in the xyz-axes is (B 0 , 0, 0) T , γ is calculated using BX, BY , B Z , α and β as sin γ = B Z sin α − BY cos α, cos γ = BX cos β .
Using the α, β, and γ values obtained, the measured magnetic flux density,B = (BX,BY ,B Z ) T , generated by the rotating magnetic dipole can be transformed into B = (Bx, By, Bz) T = RB, where Therefore, even when the sensor coordinate system is inclined relative to the global coordinate system, the outputs of the threeaxis accelerometers and three-axis magnetic sensors can be used to obtain B when ψ, ϕ, and r are estimated from Eqs. (10), (11), and (13), respectively. Two experiments were performed. In experiment 1, the sensor coordinate system coincided with the global coordinate system, but in experiment 2, the systems did not coincide. In experiment 2, the measured magnetic flux density was transformed using the rotation matrix in Eq. (16) before localization. In both experiments, the marker was located at one of the positions marked by asterisks in Figs. 4 and 5 where z = 350 mm.

A. Experimental setup
We used three magnetic impedance (MI) sensors (MI-CB-1DH, Aichi Micro Intelligent Corp.) for a sensor unit as shown in Fig. 3. These sensors were then connected to an oscillator (MI-CB-1DH-OSC, Aichi Micro Intelligent Corp.) to enable synchronous measurement. The sensor outputs were measured using an analogto-digital converter (NI9215, National Instruments) with a sampling rate of 2 kHz. The FFT of the measured data for 1 s was then computed. The detected peak frequency was 4 Hz (T = 0.25 s). Using NT = 1, Eqs. (6), (7) and (9)  angles were estimated from Eqs. (10) and (11), respectively. The average and standard deviation of the errors were evaluated for ten trials.
When the sensor was located close to the marker and the amplitude of the sensor signal exceeded the maximum or minimum output value of the MI sensor, the signal amplitude was estimated as follows. A zero crossover point for the signal was found and was set to be at 0 s. Then, by denoting the signals at time ti by di (i = 1, . . ., M), which are not saturated around the zero crossover point, the signal amplitude A and the constant, c, are estimated by solving the linear least-squares problem ∑ M i=1 |di − (A sin(ωti) + c)| 2 → min. To measure the gravitational acceleration and the geomagnetism, we used a tri-axial accelerometer and a geomagnetic sensor contained within an Android smartphone (SO-04J, Sony Mobile Communications Inc.). The degree of tilt was measured via the smartphone, which is attached to the MI sensor, using MATLAB Mobile. 17 The sampling rate was fixed at 10 Hz and the average values for 30 s were used for the transformation, as mentioned in section IIC.
For validation of the results, the magnetic marker position was also measured using an optical tracking system (Flex 3, Optitrack). The corresponding results are shown as 'reference' in Figs. 4 and 5.  In Fig. 4, the marker locations were estimated accurately and stably in the 4000 mm by 2000 mm domains without communicating any information with regard to the temporal change in the magnetic moment direction.

B. Results and discussion
In Fig. 5, the two-dimensional errors when x > 0 were larger than those obtained when x < 0. In experiment 2, the sensor coordinate was rotated toward the z-axis of the global coordinate and the marker approached the XY-plane of the sensor coordinates. As a result, |B Z | was smaller than |BX| and |BY |, and the estimated ψ and the three-dimensional position obtained were less accurate than those in experiment 1. However, the xand y-coordinates were estimated well, as shown in Fig. 5, by transforming the measured magnetic field as described in section IIC. This would therefore be sufficient for localization of the magnetic marker that moves in the xy-plane.
In experiments 1 and 2, (x * , y * , z * ) = (0, 0, 350) mm was excluded from the estimation points because | ∫ NT 0 Bze iωt dt|, which is the quantity in the denominator in Eq. (10), is equal to zero when ψ is zero. However, based on the nature of the situation where the amplitude of Bx is equal to that of By when ψ = 0, we can detect ψ = 0 as follows. First, we detect whether | ∫

IV. CONCLUSION
In this paper, we have proposed a method to estimate the position of a magnetic marker with a rotating magnetic moment. We derived a formula to estimate the azimuth and zenith angles and the distance of the marker from the sensor in terms of the Fourier components at the rotating angular frequency of the magnetic flux density, without knowledge of the direction of the magnetic moment. The position of the marker was estimated with a mean absolute error of 49.9 mm within a 4000 mm by 2000 mm domain, where z = 350 mm. Even when the sensor coordinate system did not coincide with the global coordinate system, the position of the marker was estimated using the acceleration data of the sensor with a mean absolute error of 129.4 mm. Therefore, the proposed method can be used in various indoor localization applications.