An Improved Magnetic Tracking Method using Rotating ... - IEEE Xplore

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JSEN.2014.2329186, IEEE Sensors Journal 1

An Improved Magnetic Tracking Method using Rotating Uniaxial Coil with Sparse Points and Closed Form Analytic Solution Shuang Song, Hongliang Ren and Haoyong Yu

Abstract—A set of triaxial magnetic transmitting coils fed with low frequency sinusoidal signals can be tracked with a rotating uniaxial sensing coil. This is because the specific output signal of the sensing coil will reach a maximum value when the direction of its induced magnetic field points to the transmitting coils. Usually a rotating method is used to find the maximum output direction based on the output signals from the sensing coil. This method needs a horizontal rotation phase and a vertical rotation phase to search for the maximum output and its accuracy depends on the rotation step size. In order to simplify the rotation process while maintaining the accuracy without reducing the rotation step size, an improved magnetic tracking method is proposed in this paper. It is found that when the sensing coil rotates in the horizontal plane, the specific output signals compose a sinusoidal curve. By extracting the amplitude, phase, and DC component of this sinusoidal signal, the distance and direction information between the sensing coil and transmitting coils can be estimated and no vertical rotating phase is needed. Another advantage of this method is that the tracking accuracy does not depend on the rotation step size once the sampling rate fulfills the NyquistShannon sampling theorem. Therefore, we can use sparse points to reconstruct the sinusoidal signal. Experimental results will show the effectiveness of this tracking method. Index Terms—Magnetic Tracking, Sparse Points, Closed Form Analytic Solution.

I. I NTRODUCTION

E

LECTROMAGNETIC tracking (EMT) technology has been widely used in many areas, such as human-machine interaction [1], vehicle guidance [2], robotic surgery [3]–[9], and image guided interventions [10], [11], among others. This method is mainly based on the accurate mapping of the magnetic field generated by magnetic sources [12]. Based on the magnetic sources, there are two types of magnetic tracking: permanent magnet based tracking [13]–[20] and the quasi-static electromagnetic coils based tracking [3], [21]– [29]. Compared with permanent magnet based tracking, the electromagnetic (EM) method has the advantages of antiinterference and a larger working space. Orthogonal coils are the mostly used tools to serve as the transmitting coils and This work is supported by Singapore Academic Research Fund, under Grant R397000139133, R397000157112, R397000156112, R397000173133 and NUS Grant C397000043511. Corresponding Authors: H. Ren (email: [email protected]) and H. Yu (email: [email protected]) All authors are with the Department of Biomedical Engineering, National University of Singapore,Singapore. Copyright (c) 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].

Fig. 1. Electromagnetic tracking method. A set of triaxial orthogonal transmitting coils is often served as the magnetic field source, which is sensed by a set of triaxial orthogonal sensing coils. The relative position and orientation parameters between the transmitting coils and the sensing coils can be estimated with the sensing signals.

the sensing coils in the EMT methods as shown in Fig.1. Two different estimation methods can be used in EMT, one of which is direct estimation and the other being rotation based estimation. The direct estimation method uses a set of electromagnetic coils as the transmitting source and a couple of sensing coils as the tracking target. The transmitting coils are excited with low frequency alternating current (AC) signals. By measuring the sensing signals, the positional and orientational parameters of the target can be calculated directly with an appropriate algorithm. Raad et al. [30] proposed a tracking method using both a set of triaxial magnetic dipole source coils and triaxial sensing coils. By determining small changes in the coordinates, this system can track the three dimensional position and three dimensional orientation parameters of the sensing coils. Hu et al. [22] presented an algorithm based on three-axis generating coils and three-axis sensing coils. The three dimensional position and three dimensional orientation information of the sensor coils can be determined by analytic equations directly. Li et al. [24] used eight magnetizing coils to locate an endoscopic robot in the gastrointestinal (GI) tract. These coils are stimulated to generate a quasi-static magnetic field. Plotkin et al. [26] proposed an eye tracking method by using a coplanar array consisting of eight transmitting coils. Sclera search coils (SSC) are used to implement the tracking method with the Levenberg-Marquardt (LM) algorithm. Ren et al. [3], [29] proposed a magnetic navigation system integrated with an inertial sensor to improve the accuracy of the attitude estimation in the application that with small perturbations.

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Fig. 2. Tracking method using uniaxial sensing coil and mutually orthogonal triaxial transmitting coils. The excitation signals on triaxial transmitting coils are I1 sin(ω1 t), I2 sin(ω2 t) and I3 sin(ω3 t) respectively. Coordinate system XY Z is built based on the uniaxial sensing coil. Position of the transmitting coils is (x, y, z). (m, n, p) is the direction of the sensing coil when it is rotating. θ is the angle between (x, y, z) and (m, n, p). By measuring the specific output signal A2 = A21 + A22 + A23 , distance r and direction between transmitting coils and sensing coil can be estimated with an appropriate rotation strategy.

Although the direct estimation method is easy to set up, it has the disadvantage of intensive computation. The rotation based estimation method uses rotating transmitting coil(s) to track the sensing coils. The rotation process can be carried out either mechanically or electrically. By rotating the transmitting coil(s) in a particular pattern, the three dimensional position and three dimensional orientation parameters of the sensing coils can be estimated from the sensing signals. Kuipers et al. [25] proposed a tracking method using a nutating magnetic field about a directional vector. Three-axis coils, which are excited with sinusoidal signals and a direct current (DC) signal, are used to generate the nutating magnetic field. Paperno et al. [23] used two-axis coils fed with the phase quadrature current are served as the magnetic field source, which can be treated as a rotating magnetic dipole. Determination of the position is based on measuring the phase and maximum and minimum values of the squared total of the AC part of the sensing signal. A similar method was also proposed by Song et al. [28], in which only the amplitude and phase information of the sensing signal are needed. An EMT method using rotating orthogonal coils is proposed by Ge et al. [21]. In this method, two cross-shaped coils rotating together are stimulated sequentially to generate a rotating magnetic field to track the magnetic sensors. Compared with the direct estimate method, the rotating method is complex to implement because of the rotations. On the other hand, this method has the advantage of a low computation overhead. Although the EMT method has been widely used, there are still two limitations: computational cost and signal synchronization. These disadvantages impose a limitation on the number of objects that can be tracked. In our previous work [27], [31], we presented a tracking method based on a rotating uniaxial sensing coil and mutually orthogonal triaxial transmitting coils fed with sinusoidal signals. The transmitting coils can be tracked with the uniaxial sensing coil. This is because the specific output signal of the sensing coil will reach

a maximum value when it points to the transmitting coils. As shown in Fig. 2, the distance and direction of the sensing coil can be estimated with sensing signals when the uniaxial coil is rotating in a particular pattern, which is based on the horizontal rotation phase and vertical rotation phase. This method has a low computational cost and no synchronizing signal is needed. As the sensing coil only receives the signal, adding the number of trackers can be easily done by increasing the number of tracking sensing coils. However, this method still has its limitations: For example, two rotation phases are needed to find the maximum output and the rotation step size needs to be small. In order to further simplify the rotation process and maintain the accuracy without reducing the step size, we propose an improved magnetic tracking method in this paper. It is found that when the sensing coil rotates in the horizontal plane, the specific output signals compose a cosine curve. By extracting the amplitude, phase and DC component of this cosine signal, the distance and direction between sensing coil and transmitting coils can be estimated and no vertical rotating phase is needed. Another advantage of this method is that the tracking accuracy does not depend on the rotation step size as long as the sampling rate meets Nyquist-Shannon sampling theorem. We can use limited sparse points to reconstruct the cosine curve. Thus during the rotation, no maximum output signal needs to be measured, which can lead to higher accuracy and faster speed. The primary contributions of our work are summarized as follows: • •

A fast electromagnetic tracking method with low computation cost and closed form analytic solution. Compared with the previous work [27], the proposed method only needs horizontal rotation phase to track the transmitting coils. And limited sparse points are sufficient to estimate the distance and direction information as long as the sampling rate meets Nyquist-Shannon sampling theorem.

The rest of this paper is organized as follows. First we will present the principle of the magnetic tracking method in detail in Section II. The improved method will be discussed in Section III. The simulation and experimental results will be presented in Section IV and Section V, respectively. This will be followed by the conclusion. II. P RINCIPLE OF M AGNETIC T RACKING M ETHOD A. Principle As shown in Fig. 2, the excitation signals on the triaxial transmitting coils are I1 sin(ω1t), I2 sin(ω2t) and I3 sin(ω3t) respectively. A coordinate system XY Z is built based on the uniaxial sensing coil. The position of the transmitting coils is defined as (x, y, z). (m, n, p) is the direction of the sensing coil when it is rotating. (m, n, p) is a unit vector and m2 + n2 + p2 = 1. θ is the angle between (x, y, z) and (m, n, p). Sensing signals from the sensing coil related to the three transmitting coils are A1 sin(ω1t), A2 sin(ω2t) and A3 sin(ω3t) respectively. Define A2 as



A2 = A21 + A22 + A23

(1)

When the transmitting coil is at a certain position, relationship between distance r and A2 is as follows: [27] A2 =

K 3cos2 (θ ) + 1 6 r

(2)

where K is a constant value and θ is the angle between (x, y, z) and (m, n, p) as shown in Fig. 2. It can be seen that when the uniaxial sensing coil points to the transmitting coils (that is, θ = 0 or π), the output A2 will reach a maximum when the sensing coil is at a certain position. The distance r can be decided with this maximum A2max s r=

6

4K A2max

(3)

Equation (3) shows the principle of the tracking method. We can find the maximum output A2 by using an appropriate searching strategy and subsequently calculate the distance.

B. Application This method can be used in the situation where many users need to track one target. The user can be human or robot or any other intelligent agent. Transmitting coils are mounted on the moving target and can be tracked by a sensing coil. As the sensing coil only requires receiving a signal to track the target, the adding of users can be easily done by increasing the number of tracking sensing coils. As shown in Fig. 3, XY Z is the tracking coordinate system defined by the uniaxial sensing coil. Rotation angles are denoted as α and β , where α is the angle with which the sensing coil rotates about the Z-axis and β is the angle with which the sensing coil rotates towards the Z-axis. The sensing coil first rotates about the Z-axis (defined as the horizontal rotation phase) and then rotates towards the Z-axis (defined as the vertical rotation phase). At the original state, the direction of the sensing coil is the same as the X-axis. For each rotation phase, we find the maximum output, and the final angles at each maximum output are αmax and βmax respectively. After that, the direction of the sensing coil is pointing to the transmitting coils. The distance r can then be estimated using (3) and the position of the transmitting coils can be decided as   x = rcos(βmax )cos(αmax ) y = rcos(βmax )sin(αmax )   z = rsin(βmax )

Fig. 3. Tracking method and the rotation process. XY Z is the tracking coordinate system defined by the uniaxial sensing coil. The uniaxial sensing coil is represented by the blue arrow. At the original state, the direction of the sensing coil is the same as the X-axis. First, at horizontal rotation phase, the sensing coil rotates about the Z-axis to find a maximum output, and the final angle with maximum output is αmax ; then at the vertical rotation phase, the sensing coil rotates vertically towards the Z-axis to reach a maximum output, and the final angle with maximum output is βmax . After that, the direction of the sensing coil is pointing to the transmitting coils. Then the distance r can then be estimated.

III. I MPROVED M ETHOD The previous tracking method needs horizontal and vertical rotation phases, and maximum output signal needs to be measured [27]. This may lead to a low searching speed and small rotating steps are necessary to achieve high accuracy. In order to mitigate these disadvantages, an improved method is needed to simplify the rotation process. The improved method will be presented in detail in the following part. A. Principle As shown in Fig. 4, as BA⊥OA, PA⊥OA and PB⊥OB   cosθ = OA/OP (5) cos(α − αmax ) = OA/OB   cos(βmax ) = OB/OP Then it can be seen that the relationship between θ , α, βmax and αmax is as follows: cosθ = cos(α − αmax )cos(βmax )

(4)

where (x, y, z) is the position parameter of the transmitting coils in coordinate system XY Z.

(6)

During the horizontal rotation phase, (2) can be rewritten as follows: K (7) A2 = 6 3cos2 (βmax )cos2 (α − αmax ) + 1 r Expand (7) to the following equation:



The position information can be calculated with (4) based on the distance r and the rotation angles αmax and βmax . From above, it can be seen that only the horizontal rotation phase is needed and the rotation step size can be large as long as the sampling rate meets the Nyquist-Shannon sampling theorem. Note that position parameters of the transmitting coils have nothing to do with its orientation information. That means, no matter how the transmitting coils rotate, the above results are still the same. B. Function Fitting Method with Sparse Points

Fig. 4. Relationship between α, αmax , βmax and θ . The blue arrow represents the rotating uniaxial sensing coil. α is the rotation angle of the sensing coil on the horizontal plane (XY plane) about Z-aixs. αmax represents the rotation angles that with the maximum output A2 during the horizontal rotation phase. βmax is the rotation angle that towards Z-axis with the maximum output A2 during the vertical rotation phase. P is the position of the transmitting coils, PB⊥XOY and BA⊥OA, so we can see that PA⊥OA.

According to the Nyquist-Shannon Sampling Theorem, we can only sample limited sparse points to reconstruct the cosine signal. There are many methods to reconstruct the cosine signal and extract the amplitude, phase and DC information of the cosine signal, such as FFT and function fitting method (FFM) [32], [33]. Here we will use FFM, which is based on the Least Square Method. Rewrite (9) as follows: π + ϕ) + c 2 = asin(ωαi + ϕs ) + c

A2i = asin(ωαi +

(14)

= as sin(ωαi ) + ac cos(ωαi ) + c where

3K 3K K cos2 (βmax )cos(2α − 2αmax ) + 6 cos2 (βmax ) + 6 6 2r 2r r (8) 2 We can see that A is a cosine function of α. Defining the cosine function as follows,

 π  ϕs = 2 + ϕ 2 as + a2c = a2   tanϕs = aacs

A2 =

A2 = acos(ωα + ϕ) + c

(9)

where a is the amplitude of the AC part, ϕ is the phase of the AC part, ω is related to the frequency of the AC part and c is the DC part. Comparing (9) with (8), it can be seen that  3K 2 a = 2r  6 cos (βmax )   ω = 2 (10)  ϕ = −2αmax    3K K 2 c = 2r 6 cos (βmax ) + r6 By applying an appropriate extraction method which will be introduced in the following part, the amplitude a, phase ϕ and DC part c can be estimated. The distance r and rotation angles αmax and βmax can then be calculated according to (10) s K (11) r= 6 c−a αmax = − s βmax = arccos

ϕ 2

(12)

2a 3(c − a)

(13)

(15)

αi = (i − 1)γ, γ is the rotation step size. i(= 1...N) indicates the i-th sample during the rotation and there will be N sample data that are used to fit this sinusoidal function. as , ac and c can be calculated as follows by using FFM and the least square method (LSM) N

Err = ∑ (Vi − A2i )2

(16)

i=1

where Vi represents the i-th sampling value of the sparse points. Minimize Err  ∂ Err   ∂ as = 0 ∂ Err (17) ∂ ac = 0   ∂ Err ∂c = 0 and then these three parameters can be calculated from the following equations:   N V sin(ωα ) ∑ i   i=1 i     as N   ac  = M−1  ∑ Vi cos(ωαi ) (18) i=1    c N   ∑ Vi i=1

where



Fig. 5. Rotation and Output. The uniaxial sensing coil rotates on the horizontal plane (XY plane). Position of the target is (x, y, z), where x > 0, y > 0 and z > 0. Each red star in the right figure represents an output signal with the rotation status in the left figure. Here the rotation step size is set to be π6 . Parameters in (9) are assumed as: a = 1, ϕ = π4 , c = 2. These seven samples can be used to extract the amplitude, phase and DC information by using function fitting method. With amplitude, phase and DC information, position parameters can then be estimated.

M= 

N

sin2 (ωα

N

∑ i)  i=1  N  ∑ sin(ωαi )cos(ωαi ) i=1  N  ∑ sin(ωαi ) i=1



N

∑ sin(ωαi )cos(ωαi ) i=1 N

∑ cos2 (ωαi ) i=1 N

∑ cos(ωαi )

∑ sin(ωαi )    ∑ cos(ωαi )  i=1  N  ∑1

i=1

i=1 N

i=1

(19)

Based on as and ac , a and ϕ can be estimated. C. Application and Comparison The improved tracking method will be carried out as shown in Fig. 5. The uniaxial sensing coil rotates on the horizontal plane (XY plane). The position of the target is (x, y, z). Each red star in the right figure represents an output signal with the rotation status in the left figure. Here the rotation step size is set to be γ = π6 , so N = 7. Parameters in (9) are assumed as: a = 1, ϕ = π4 , c = 2. These seven samples can be used to extract the amplitude, phase and DC information by using the function fitting method. With the amplitude, phase and DC information, the position parameters can then be estimated. The algorithm flowchart of the previous method and the improved method can be seen in Fig. 6. Compared with the previous method, the improved method has simplified the searching procedure. It is also not restricted by the rotation step size as long as the sampling rate meets the NyquistShannon sampling theorem. IV. S IMULATION Before experimental validations, simulations were done to compare the proposed method with the previous method. The

Fig. 6. Algorithm flowchart of previous method and improved method. Compared with the previous method, the improved method not only simplifies the searching procedure, but also simplifies the sampling as only sparse points are needed.

simulations were performed with positions of the transmitting coils randomly distributed in the region bounded by x ∈ (0, 1)m, y ∈ (0, 1)m and z ∈ (0, 1)m. Under this assumption, α ∈ (0, π2 ) and β ∈ (0, π2 ). Orientation of the transmitting coils is also random. Six different rotation step sizes are used in the simulation. Tab. I shows the simulation results with 0.1% noise and Tab. II shows the results with 1% noise. The relationship between Step Size and Sampling Points is Ns =

90◦ +1 γ


(20)


TABLE I S IMULATION R ESULT WITH 0.1% N OISE . U NIT : MM Step Size

Sampling Points

45◦ 30◦ 18◦ 9◦ 1◦ 0.1◦

3 4 6 11 91 901

Error of Previous x y z 118.6 88.71 51.2 25.6 2.8 0.3

115 92.4 52.8 26.5 2.9 0.3

140.6 104.6 62 30.3 3.3 0.3

Error of Improved x y z 0.2 0.2 0.2 0.2 0.2 0.2

0.2 0.2 0.2 0.2 0.2 0.2

0.9 0.8 0.8 0.9 0.8 0.8

Fig. 8. Simulation result of the relationship between step size and error with noise 1%

TABLE II S IMULATION R ESULT WITH 1% N OISE . U NIT : MM Step Size

Sampling Points

45◦ 30◦ 18◦ 9◦ 1◦ 0.1◦

3 4 6 11 91 901

Error of Previous x y z 119 88.5 53.8 26.3 3.2 1.2

121.7 87.6 53 24.6 3 1.5

144.5 99.8 60.1 29.8 3.6 1.5

Error of Improved x y z 2.3 2.3 2.6 2.2 2.9 2.9

2.3 2.2 2.4 2.3 2.7 2.9

6.3 6.7 7.5 7 6.7 6.9

where Ns is the number of Sampling Points and γ represents the Step Size. From the simulation results, it can be seen that 1) when the number of sampling points is small, the improved method is much better than the previous method; 2) the accuracy of the improved method does not depend on the rotation step size, it is only affected by noise level. This conclusion can be drawn from Fig. 7 and Fig. 8. Fig. 7 shows the relationship between the rotation step size and errors of x, y, z, r with noise level of 0.1% and Fig. 8 shows the relationship with noise level of 1%. It can be seen that when the rotation step size increases from 1◦ to 45◦ , the localization error is maintained at the same level.

Fig. 9. Experimental Platform. The uniaxial sensing coil is mounted on a rotary device and a set of triaixal transmitting coils is used as the tracking target. During the experiment, for each test position, the uniaxial sensing coil rotates from 0◦ to 90◦ with a rotation step of 15◦ . That is, seven sparse points are used to reconstruct the cosine signal for each position.

V. E XPERIMENTAL R ESULTS A. Experimental Results An experimental platform has been set up to validate this tracking method. As shown in Fig. 9, a uniaxial sensing coil is mounted on a rotary device and a set of triaxial transmitting coils is used as the target. We have tested 22 positions, for each position, the uniaxial coil rotates from 0◦ to 90◦ with a rotation step of 15◦ . Fig. 10 shows the tracking result of the 22 test positions. Fig. 11 shows the distance error of the test positions. The average distance error is 2.8mm. B. Discussion

Fig. 7. Simulation result of the relationship between step size and error with noise 0.1%

Although the distance error is acceptable, the position error is not good for our experiments. The position error depends on the errors on αmax and βmax . The average errors of αmax and βmax are 7.9◦ and 4.7◦ , respectively. A small error on these two angles will lead to a big error on (x, y, z), especially when



Fig. 10. Tracking result of the 22 test positions. In this figure, blue points represent the true distance values and red points are for the estimated results.

Fig. 12. Relationship between the error of αmax and βmax and error of X. Here distance r is set to be 0.3m, αmax = 60◦ , βmax = 45◦ . Errors on αmax and βmax are from 0◦ to 5◦ .

extracting the amplitude, phase and DC part of this cosine signal, the distance and direction between sensing coil and transmitting coils can be estimated and no vertical rotating phase is needed. Another advantage of this method is that the tracking accuracy does not depend on the rotation step size as long as the sampling rate meets the Nyquist-Shannon sampling theorem. Simulation results show that this method can have good performance with large rotation step size. Experimental results show the effectiveness of this tracking method. R EFERENCES

Fig. 11. Distance Error. The average distance error is 2.8mm.

the distance r is large. The relationship between the error of αmax and βmax and error of position can be seen in Fig. 12. Here, distance r is set to be 0.3m, αmax = 60◦ , βmax = 45◦ . Errors on αmax and βmax are from 0◦ to 5◦ . XError is defined as follows: XErr = |rcos(βmax + βErr )cos(αmax + αErr ) −rcos(βmax )cos(αmax )|

(21)

The error will increase as the value of distance r increases. In the future, we will improve our experimental device to get more accurate results. VI. C ONCLUSION We proposed an improved magnetic tracking method in this paper. This method is based on triaxial transmitting coils of AC signals and rotating uniaxial sensing coils. It is found that when the sensing coil rotates in the horizontal plane, the specific output signals compose a cosine curve. By

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