Interpolation of the magnetic field at the test masses in eLISA

Since the magnetic sources in the spacecraft are located far from the origin of the coordinate system (chosen at the center of the TM) and assuming the material inside the vacuum enclosure is basically non-magnetic, the magnetic field in this region can be considered to be essentially a vacuum field (${\boldsymbol{\nabla }}\times {\bf{B}}={\boldsymbol{\nabla }}\cdot {\bf{B}}=0$). Hence, the estimated magnetic field ${\boldsymbol{}}{{\bf{B}}}_{{\rm{e}}}$ obtained employing an array of N sensors can be written as the general solution to Laplace's equation centered at the TM, which can be expressed in terms of an expansion in spherical harmonics:

$$\begin{eqnarray}&&{{\bf{B}}}_{{\rm{e}}}({\bf{x}})={\boldsymbol{\nabla }}{\rm{\Psi }}({\bf{x}})=\underset{l=1}{\overset{L}{\displaystyle \sum }}\underset{m=-l}{\overset{l}{\displaystyle \sum }}\;{M}_{{lm}}(t){\boldsymbol{\nabla }}[{r}^{l}\;{Y}_{{lm}}({\bf{n}})],\end{eqnarray} \tag{ 2 }$$

where $r\equiv | {\bf{x}}|$ and ${\bf{n\ \equiv \ x}}/r$ are the spherical coordinates of the field at ${\bf{x}}$. M_lm and Y_lm are the multipole coefficients and the standard spherical harmonics of degree l and order m, respectively [12].

The accuracy of the estimation of the magnetic field is given by the order of the expansion, which depends on the number of multipole coefficients that can be computed. Specifically, the accuracy of the interpolation is given by the number of known magnetic field measurements at the boundary of the volume where the field equations are considered. In our case these measurements are provided by the number of magnetometers placed in the spacecraft. Table 1 shows the minimum number of magnetometers required to model the magnetic field with a second, third and fourth order multipole expansion.

The coefficients M_lm are found by minimizing the equation $\partial {\varepsilon }^{2}/\partial {M}_{{lm}}=0$, where the square error is defined as

$$\begin{eqnarray}&&{\varepsilon }^{2}({M}_{{lm}})=\underset{s=1}{\overset{N}{\displaystyle \sum }}{\left|{{\bf{B}}}_{{\rm{m}}}({{\bf{x}}}_{s})-{{\bf{B}}}_{{\rm{e}}}({{\bf{x}}}_{s})\right|}^{2},\end{eqnarray} \tag{ 3 }$$

${{\bf{B}}}_{{\rm{m}}}$ is the readout of the triaxial magnetometer, and N is the total number of magnetometers. This is done employing a least-squares method. Once the system of equations is solved, the computed coefficients M_lm can be inserted into equation (2), replacing the magnetometer's position, ${{\bf{x}}}_{s}$, by the TM position, ${{\bf{x}}}_{\mathrm{TM}}$, to finally obtain the value of the interpolated field at the TM location.

The magnetic field at the TM position inferred from the readings of the magnetometers can also be approximated by a Taylor expansion. As in the case in which the multipole expansion is employed, the order of the Taylor series is determined by the number of magnetometer data channels. In this case the magnetic field at the position of the TMs can be approximated by the following expression:

$$\begin{eqnarray}&&{{\bf{B}}}_{{\rm{m}}}({{\bf{x}}}_{s})={{\bf{B}}}_{{\rm{e}}}({{\bf{x}}}_{\mathrm{TM}})+\underset{n=1}{\overset{L}{\displaystyle \sum }}\underset{i=1}{\overset{3}{\displaystyle \sum }}\displaystyle \frac{{\partial }^{n}{{\bf{B}}}_{{\rm{e}}}({{\bf{x}}}_{\mathrm{TM}})}{\partial {x}_{i}}\displaystyle \frac{{({x}_{s,i}-{x}_{\mathrm{TM},i})}^{n}}{n!},\end{eqnarray} \tag{ 4 }$$

where the origin of coordinates is defined at the center of the respective TM (${{\bf{x}}}_{\mathrm{TM}}$), and ${{\bf{x}}}_{s}$ are the magnetometer locations. ${{\bf{B}}}_{{\rm{e}}}({{\bf{x}}}_{\mathrm{TM}})$ and ${\partial }^{n}{{\bf{B}}}_{{\rm{e}}}({{\bf{x}}}_{\mathrm{TM}})/\partial {x}_{i}$ are calculated considering that the magnetic field around the TM has both zero divergence and curl, i.e. the magnetic field gradient tensor ${{\boldsymbol{\nabla }}}^{n}{\bf{B}}$ is a symmetric and traceless matrix. Thus, only a total of five independent components need to be computed.

This method consists of computing the field as a weighted sum of the different magnetometer readings. The calculation is performed as follows:

$$\begin{eqnarray}&&{{\bf{B}}}_{{\rm{e}}}=\underset{s=1}{\overset{N}{\displaystyle \sum }}{a}_{s}{{\bf{B}}}_{{\rm{m}}}({{\bf{x}}}_{s}),\end{eqnarray} \tag{ 5 }$$

where ${{\bf{B}}}_{{\rm{m}}}({{\bf{x}}}_{s})$ are the readouts of the magnetometers. The weighting factors a_s are given by:

$$\begin{eqnarray}&&{a}_{s}=\displaystyle \frac{1/{r}_{s}^{n}}{{\displaystyle \sum }_{i=1}^{N}1/{r}_{i}^{n}},\end{eqnarray} \tag{ 6 }$$

where n specifies the order of the interpolation and r_i are the distances between the point at which the field must be estimated and the specified magnetometer.

As previously explained, to validate the performance of the reconstruction algorithm, a batch of dipoles with randomly generated orientations were simulated and the exact magnetic field for each one of these realizations was compared with the interpolated results. The left panel of figure 3 shows the x-component of the magnetic field map produced by one of these random configurations. The results are then compared in the right panel with those obtained using one of our interpolating methods, in this case the magnetic field reconstructed using multipole expansion. As seen in section 2, a multipole expansion based only on eight triaxial magnetometers readings is able to resolve the magnetic field up to the hexadecapole structure, by computing 24 terms in equation (2). Overall, the field qualitatively resembles the exact one, although there are apparent differences far from the positions of the TMs. However, note that the success of the reconstruction method is determined by the accuracy achieved at the region of interest, i.e. at the TM locations. We perform a more quantitative analysis for the three components of the field below.

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The differences (in percentage) between the interpolated field and the source dipole model field are shown in figure 4. Contour plots for the three components and the modulus show the accuracy achieved by the multipole algorithm. As can be seen in this figure, the smallest differences occur in the region enclosed by the magnetometers. Moreover, the accuracy of the interpolating algorithm is good in the central area of the electrode housing, where the TM is located.

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To further confirm the validity and general applicability of the multipole expansion, we compared the differences between the interpolated and exact magnetic field at the position of the TM for three different sensor layouts. Specifically, we first adopted the LISA Pathfinder configuration. In this layout fluxgate magnetometers are used, as depicted in figure 1. In a second step we did the same, adopting four AMRs placed around the vacuum enclosure at the height of the electrode housing center. Finally, we carried out the same calculation, this time adopting eight AMRs, as graphically displayed in figure 2. Average and maximum field errors relative to the modulus (${\bar{\varepsilon }}_{| {\bf{B}}| }$ and ${\varepsilon }_{| {\bf{B}}| ,\mathrm{max}}$) and to the field components (${\bar{\varepsilon }}_{{B}_{i}}$) over the 10³ random runs are shown in table 2. In the LISA Pathfinder configuration the accuracy of the reconstructed field at the TM is poor and presents large variations when the multipole expansion is used. In particular, the estimation errors can be as high as $737\%$. This is the natural consequence of having placed the sensors too far from the center of the TM. Instead, when AMRs are used, the sensors can be placed much closer to the center of the TM, due to its smaller size and intrinsic magnetic moment. The results when the same number of magnetometers is employed show significant improvements, with maximum errors up to 15%. Finally, the estimation errors are reduced by a factor of ∼6 (${\varepsilon }_{| {\bf{B}}| ,\mathrm{max}}=2.4\%$) when eight sensors are used. In this case the hexadecapole expansion can be employed, and this obviously results in an improved performance of the interpolating algorithm. Last, in figure 5 the distribution of the estimation errors for the randomly simulated cases is shown. This figure clearly shows that the standard deviations are $\leqslant 1.1\%$ and $\leqslant 0.18\%$ for the 4-AMR and 8-AMR layout, respectively. This proves that the averaged estimation errors (${\bar{\varepsilon }}_{| {\bf{B}}| }\leqslant 0.2\%$) are robust and that the performance of the multipole interpolating algorithm is good, providing reliable estimated values of the magnetic field at the location of the TM.

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Table 2.  Relative errors of the magnetic field estimation at the positions of the TM. ${\bar{\varepsilon }}_{| {\bf{B}}| }$ and ${\bar{\varepsilon }}_{{B}_{i}}$ are the mean error for a batch of 10³ randomly orientated magnetic sources relative to the modulus $| {\bf{B}}|$ and to the field component ${B}_{i}$, respectively. The denominator in ${\bar{\varepsilon }}_{{B}_{i}}$ is closer to zero than that of the modulus ${\bar{\varepsilon }}_{| {\bf{B}}| }$, and this translates in larger errors for the x-component than for the modulus.

Error	LPF (4 fluxgates)				eLISA (4 AMRs)				eLISA (8 AMRs)
($\%$)	Bx	By	Bz	$| {\bf{B}}|$	Bx	By	Bz	$| {\bf{B}}|$	Bx	By	Bz	$| {\bf{B}}|$
${\bar{\varepsilon }}_{| {\bf{B}}| }$	38.2	28.1	20.9	32.5	1.4	1	1.1	1.8	0.1	0.2	0.1	0.1
${\varepsilon }_{| {\bf{B}}| ,\mathrm{max}}$	737.7	340.3	327.6	803.2	15.0	7.7	14.0	13.3	0.9	2.4	1.4	2.0
${\bar{\varepsilon }}_{{B}_{i}}$	697.9	202.1	184.5	32.5	13.7	3.8	7.8	1.8	0.6	0.8	5.3	0.1

The results described so far were obtained using the multipole expansion algorithm. However, other interpolation schemes were detailed in section 2, and their performance compared with that of the multipole expansion in table 3. The order of the interpolation in the distance weighting method is set to n = 1. Nevertheless, this choice is not relevant due to the physical symmetry of the sensor placement, i.e., the distances r_s, and consequently the weighting factors a_s, are equivalent for the eight magnetometers. For the Taylor expansion, the second and the terms involving higher-order derivatives are negligible due to the symmetry of the magnetic distribution. Thus, the Taylor approach mainly estimates the magnetic field as a linear approximation. For this reason, we expect the results of the interpolation to be almost identical to those obtained using the distance weighting method. Table 3 shows the accuracies of the estimation of the magnetic field at the position of the TM for the three methods employed in this work. As can be seen, the multipole expansion outperforms by far the rest of the methods described previously.

Table 3.  Maximum errors of the estimated magnetic field at the position of the TM using different interpolation methods, see text for details.

	${\varepsilon }_{| {\bf{B}}| ,\mathrm{max}}\;[\%]$
Error	Bx	By	Bz	$| {\bf{B}}|$
Distance weighting	8.0	4.0	7.7	7.9
Taylor expansion	8.0	4.0	7.7	7.9
Multipole expansion	0.9	2.4	1.4	2.0

Magnetic field gradients also need to be estimated from the readouts of the eight AMRs. We do this using the multipole expansion algorithm because, as demonstrated earlier, this interpolating method outperforms the other two methods studied here. For the sake of clarity, only the errors of the gradient interpolation for two components ($\partial {B}_{x}/\partial x$ and $\partial {B}_{z}/\partial x$, respectively) along the spacecraft are shown in figure 6. In this case, minimum errors are also obtained in the center of the TM, though unlike that obtained for the case of the magnetic field, the error increases somewhat faster in the region outside of the boundary of the area surrounding the magnetometers. Additionally, relative errors around the TM area are slightly larger than those found for the reconstruction of the magnetic field, although they remain lower than 3%. Figure 7 shows the distribution of the estimation errors and standard deviations for five independent components of the gradient matrix ${\boldsymbol{\nabla }}{\bf{B}}$ at the position of the TM. Inspection of this figure reveals that the multipole expansion scheme is robust. In particular, when this interpolant is used, we obtain not only accurate values of the reconstructed magnetic field, but also of its gradient, with typical accuracies of the order of 2%, and deviations below $2.5\%$ respectively.

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Absolute errors and drifts of the magnetometers readings are relevant to the interpolation quality, since the algorithm is entirely based on the magnetometer outputs. Due to the stringent stability requirements for eLISA, drifts of the measurements are not critical. Thus, the analysis is focused on the absolute errors. To validate the robustness of the system, the performance of the multipole expansion scheme is studied for two common sources of error. Namely, possible offsets in the magnetometer readings and spatial uncertainty—that is, deviations from the nominal position of the sensor core. We analyze their eventual effects separately. Offsets in the magnetometer or in the signal conditioning circuit can be measured on-ground and considered in the analysis. However, unknown magnetometer offsets due to launch stresses can lead to inaccurate field determination [19]. The precision of the position of the sensors may eventually be another source of error that cannot be ignored 'a priori'. The spatial uncertainty depends on the size of the sensor head, since smaller heads result in a smaller uncertainty of the precise location of the measurement.

The offsets of the magnetometers can be relevant depending on the measurement technique. In particular, for AMR sensors, flipping signals applied to the sensor help to overcome the offset by reversing the sensor magnetization and modulating the output signal [20]. The changes in the direction of the sensor magnetization lead to inversion of the output characteristics but not the offset, which can be canceled by subtracting the measurements between each flipping pulse. Regarding the spatial uncertainty, the layout of the thin film forming the AMR Wheatstone bridge [21] is deposited by a sputtering process, and has a rough area of $0.9\times 1.2\ {\mathrm{mm}}^{2}$. Therefore, a spatial uncertainty smaller than $1\;\mathrm{mm}$ is expected.

The impact of these effects on the accuracy of the multipole expansion algorithm is simulated as follows. First, a $3\times N$ matrix of offsets is randomly generated according to a uniform distribution with an interval of $[-{B}_{\mathrm{Offset}},{B}_{\mathrm{Offset}}]$. Second, the offset array is added to the $3\times N$ magnetic channels readings, and finally the magnetic field and errors are estimated. These steps are sequentially repeated for series of 10³ random offsets with intervals of the same length. A similar procedure is done to assess the robustness of the interpolation to the uncertainty in the location of the sensor heads. The maximum estimation errors as a function of the offset and of the spatial uncertainty are shown in figure 8. As can be observed, the offset of the sensor is more determinant than its spatial resolution. Specifically, for an unpredictable non-measured offset of $10\;\mathrm{nT}$, the maximum estimation error is $\sim 42\%$. These results reflect the relevance of the magnetic sensing technology. Specifically, we stress that appropriate techniques to cancel out the undetermined offset and the use of tiny sensors with accurate spatial resolution are totally necessary.

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