# Calibration of the in-orbit center-of-mass of TaiJi-1

Xiaotong Wei<EMAIL_ADDRESS>University of Chinese Academy of
Sciences (UCAS), Beijing 100049, China  International Centre for Theoretical
Physics Asia-Pacific, UCAS, Beijing 100190, China  Taiji Laboratory for
Gravitational Wave Universe (Beijing/Hangzhou), UCAS, Beijing 100190, China
Li Huang  University of Chinese Academy of Sciences (UCAS), Beijing 100049,
China  International Centre for Theoretical Physics Asia-Pacific, UCAS,
Beijing 100190, China  Taiji Laboratory for Gravitational Wave Universe
(Beijing/Hangzhou), UCAS, Beijing 100190, China  Tingyang Shen  University of
Chinese Academy of Sciences (UCAS), Beijing 100049, China  International
Centre for Theoretical Physics Asia-Pacific, UCAS, Beijing 100190, China
Taiji Laboratory for Gravitational Wave Universe (Beijing/Hangzhou), UCAS,
Beijing 100190, China  Zhiming Cai  Innovation Academy for Microsatellites,
Chinese Academy of Sciences, Shanghai 201304, China  Jibo He
<EMAIL_ADDRESS>University of Chinese Academy of Sciences (UCAS), Beijing
100049, China  International Centre for Theoretical Physics Asia-Pacific,
UCAS, Beijing 100190, China  Taiji Laboratory for Gravitational Wave Universe
(Beijing/Hangzhou), UCAS, Beijing 100190, China  Hangzhou Institute for
Advanced Study, UCAS, Hangzhou 310024, China

###### Abstract

Taiji program is a space mission aiming to detect gravitational waves in the
low frequency band. Taiji-1 is the first technology demonstration satellite of
the Taiji Program in Space, with the gravitational reference sensor (GRS)
serving as one of its key scientific payloads. For accurate accelerometer
measurements, the test-mass center of the GRS must be positioned precisely at
the center of gravity of the satellite to avoid measurement disturbances
caused by angular acceleration and gradient. Due to installation and
measurement errors, fuel consumption during in-flight phase, and other
factors, the offset between the test-mass center and the center-of-mass (COM)
of the satellite can be significant, degrading the measurement accuracy of the
accelerometer. Therefore, the offset needs to be estimated and controlled
within the required range by the center-of-mass adjustment mechanism during
the satellite’s lifetime. In this paper, we present a novel method, the
Extended Kalman Filter combined with Rauch-Tung-Striebel Smoother, to estimate
the offset, while utilizing the chi-square test to eliminate outliers.
Additionally, the nonlinear Least Squares estimation algorithm is employed as
a crosscheck to estimate the offset of COM. The two methods are shown to give
consistent results, with the offset estimated to be $dx\approx$$-$$0.19$ mm,
$dy\approx 0.64$ mm, and $dz\approx$$-$$0.82$ mm. The results indicate a
significant improvement on the noise level of GRS after the COM calibration,
which will be of great help for the future Taiji program.

††preprint: APS/123-QED

##  I Introduction

In 2016, the LIGO collaboration made a groundbreaking discovery by detecting
gravitational waves (GW) [1], thereby providing a direct verification of the
predictions made by Albert Einstein in his general theory of relativity a
century ago [2]. This discovery has had a profound impact on basic scientific
research worldwide.

Space-based gravitational wave detection represents an intriguing frontier for
future studies of the gravitational universe, as it can extend the reach of
gravitational wave astronomy beyond that of the ground-based detectors,
thereby allowing for a wider range of gravitational radiation sources to be
observed [3], which will provide invaluable information to deepen our
understanding of the evolution of early universe and the nature of gravity.
Several space-borne gravitational-wave observatories have been proposed, such
as LISA [4, 5, 6], DECIGO [7], ASTROD [8], Taiji [3, 9], and Tianqin [10].

The Taiji program, initiated by the Chinese Academy of Sciences (CAS), is a
space mission that aims to detect gravitational waves (GW) in the frequency
band between 0.1 mHz and 1.0 Hz, which is important in the fields of astronomy
and cosmology [9, 11, 12]. The Taiji program proposes to detect GW signals
using the Michelson laser interferometer principle, where each end of the
interferometer contains a test mass (TM) serving as a reference body. This
reference body is required to be free from spurious accelerations relative to
its local inertial frame, and any spurious accelerations will affect the
detection of tidal deformations caused by gravitational waves.

To facilitate the development of technology for the Taiji program, a three-
step road map has been proposed [13, 14]. As the first step, a technology
demonstrator satellite, Taiji-1 [14, 15], was launched on August 31, 2019. One
of the key technologies validated by Taiji-1 is the Gravity Reference Sensor
(GRS), which serves as an accelerometer and consists of sensors and electronic
components [16]. The sensor comprises an electrode housing and a TM surrounded
by the sensing electrode, as shown in Fig. 1.

GRS has three axes, including one non-sensitive axis and two sensitive axes,
the directions of $+\textrm{X}$, $+\textrm{Z}$, and $+\textrm{Y}$ in Fig. 1
correspond to the non-sensitive axis (radial direction), the first sensitive
axis (flight direction), and the second sensitive axis, respectively. The
sensor utilizes capacitive sensing technology to measure the disturbance
acceleration of Taiji-1. The resulting data is sent to the drag-free
controller, which instructs the thruster to apply force to compensate for the
disturbing force. As the reference body of the future Taiji program
interferometer, GRS needs to effectively mitigate the impact of all non-
gravitational accelerations so that to reach the desired sensitivity of Taiji.
Moreover, the accuracy of GRS is crucial for the drag-free control system, as
the GRS readout results are used as inputs for issuing commands to control the
spacecraft. GRS is susceptible to a range of noise sources, including Brownian
noise arising from the surrounding air near TM, charge-induced noise due to TM
charge accumulation, readout noise from voltage signals, temperature
gradients, circuit noise, magnetic noise, self-gravity noise.

A precise positioning of the test-mass of the accelerometer at the center-of-
mass (COM) of the Taiji-1 satellite is crucial to suppress non-gravitational
and perturbed accelerations such as angular motion-related accelerations and
accelerations due to gravity gradients [17, 18]. The COM of the accelerometer
is adjusted to be at the COM of the satellite before launching. However,
during the satellite’s operation, the consumption of propellant causes the COM
of the satellite to change relative to the satellite frame, leading to a shift
of the accelerometer COM relative to the satellite COM over time. Therefore,
it is crucial to regularly measure the deviation of the COM position of the
two during the entire life cycle of the satellite, and use the COM adjustment
mechanism to perform in-orbit adjustments so that the deviation is within a
certain range [19, 20, 21].

In Section II, the principle of the COM calibration is presented. The
accelerometer measurement model is described in Section III. We discuss the
use of the Extended Kalman filter model and Rauch-Tung-Striebel Smoother for
COM calibration in Section IV, while the detection and removal of outliers
principle is explained in Section V. The performance of the COM calibration is
evaluated in Section VI, and the results and conclusions are summarized in
Section VII.

##  II Principle of COM calibration during in-orbit

As shown in Fig. 1, the primary components of the high-precision electrostatic
levitation accelerometer include a free fall TM, a set of capacitive electrode
plate that surrounds the TM (together forming a sensitive probe), and a
peripheral capacitive sensing and electrostatic feedback control circuit. This
circuit enables the detection of position and attitude changes between the TM
and the electrodes, as well as the measurement of acceleration through its
feedback voltage. There are six sensing, control, and feedback circuits that
use the same principle to measure three translational accelerations and three
angular accelerations of the TM concurrently.

An offset of the COM of the electrostatic accelerometer from the COM of the
satellite can cause measurement disturbances, primarily attributed to the
angular motion of the satellite. Therefore, an estimate of the offset can be
achieved using appropriate algorithms based on the relationship between the
angular motion and the measurement disturbance. In the calibration experiment,
a signal of certain frequency is injected into the attitude control to induce
magnetic torque of satellite, which will cause disturbance to the
accelerometer measurements and ensure its magnitude is sufficiently large to
disregard other disturbance effect, such as solar pressure torque and
aerodynamic torque. Then, the offset can be modeled from readout of
accelerometer and star tracker.

Figure 1: The geometric structure of the core mechanical assembly of the GRS
[22].

##  III Accelerometer measurement Model

The accelerometer measurement output model of the relative acceleration of TM
and the electrode cage for Taiji-1 is presented as follows [15, 17, 18],

$a_{\textrm{out}}=\ddot{d}+\dot{\omega}\times
d+2\omega\times\dot{d}+\omega\times(\omega\times
d)+a_{\textrm{g}}+a_{\textrm{ng}}$ (1)

where $a_{\textrm{out}}$ represents the theoretical measurements of the
accelerometer, $d$ represent the COM offset between the accelerometer and the
satellite, $\dot{d}$ and $\ddot{d}$ denote the first and second partial
derivation of $d$ relative to time. $\omega$ and $\dot{\omega}$ are the
angular velocity and the angular acceleration respectively. $a_{\textrm{g}}$
is the acceleration due to the gravitational gradient, the accelerometer
measurement model ignores the perturbations due to the solid Earth tides,
ocean tides, rotational deformations, the planets including the Sun and Moon,
and general relativity which can be neglected as for the calibration span is
short enough. Furthermore, $a_{\textrm{ng}}$ represents the non-gravitational
acceleration on the satellite, such as atmospheric drag, solar radiation
pressure, and the Earth radiation pressure.

During the offset calibration period, the deviation of the COM of the TM from
the COM of the satellite can be approximated as a constant offset, given the
short measurement time. Therefore, $a_{\textrm{out}}$ can be expressed as,

$a_{\textrm{out}}=\dot{\omega}\times d+\omega\times(\omega\times
d)+a_{\textrm{g}}+a_{\textrm{ng}}.$ (2)

Considering the relatively smooth change of acceleration caused by non-
conservative forces and gravity gradient, it can be approximated as a linear
change within a limited time span. Therefore, the calibration interval is
designed to be several minutes.

The attitude control system of the Taiji-1 satellite is equipped with a star
tracker and a gyroscope to respectively measure the attitude angle of the
satellite relative to the inertial coordinate system and the angular velocity
$\omega$. Additionally, the angular accelerations $\dot{\omega}$ are obtained
by a second-order polynomial fitting method.

Assuming negligible scale factors and misalignment errors, after substituting
the $\omega$ and $\dot{\omega}$, the accelerometer measurement model can be
expressed as follows,

$A_{\textrm{out}}=\widetilde{A}d+\alpha t+\beta+A_{\textrm{n}},$ (3)

where $\tilde{A}$ can be expressed as

$\tilde{A}=\left[\begin{array}[]{ccc}-\omega_{\textrm{y}}^{2}-\omega_{\textrm{z}}^{2}&\omega_{\textrm{x}}\omega_{\textrm{y}}-\dot{\omega}_{\textrm{z}}&\omega_{\textrm{x}}\omega_{\textrm{z}}+\dot{\omega}_{\textrm{y}}\\\
\omega_{\textrm{x}}\omega_{\textrm{y}}+\dot{\omega}_{\textrm{z}}&-\omega_{\textrm{x}}^{2}-\omega_{\textrm{z}}^{2}&\omega_{\textrm{z}}\omega_{\textrm{y}}-\dot{\omega}_{\textrm{x}}\\\
\omega_{\textrm{x}}\omega_{\textrm{z}}-\dot{\omega}_{\textrm{y}}&\omega_{\textrm{z}}\omega_{\textrm{y}}+\dot{\omega}_{\textrm{x}}&-\omega_{\textrm{y}}^{2}-\omega_{\textrm{x}}^{2}\end{array}\right].$
(4)

The corresponding axis of the spacecraft can be represented by
$\omega_{\textrm{i}}$ and $\dot{\omega}_{\textrm{i}}$
($\textrm{i}=\textrm{x},\textrm{y},\textrm{z}$) for angular acceleration and
angular velocity, respectively. The linear slope is represented by $\alpha$
and the constant bias by $\beta$. In this study, we removed the linear effect
by detrending $A_{\textrm{out}}$. Therefore, the model of $A_{\textrm{out}}$
utilized in this study is,

$A_{\rm out}=\widetilde{A}d+A_{\textrm{n}}$ (5)

where $A_{\textrm{n}}$ is the measurement noise diagonal matrix.

##  IV Extended Kalman filter model and Rauch-Tung-Striebel Smoother for COM
calibration

Kalman filter is a high-efficiency recursive filter [23]. The filtering theory
proposed by Kalman is only applicable to linear systems. An Extended Kalman
Filter (EKF) was proposed in Ref. [24, 25], and can be applied to the
nonlinear field.

The accelerometer measurement model used in this study is given by Eq. 5, and
one can define the state variable as the COM offset and use the Kalman filter
to estimate the offset. Here, the state vector is denoted as
$X=[d_{\textrm{x}},d_{\textrm{y}},d_{\textrm{z}}]$ with
$\dot{d_{\textrm{i}}}=0$ for $\textrm{i}=\textrm{x},\textrm{y},\textrm{z}$.
The state equation, as derived in Ref. [20], is,

$\hat{X}_{k}=\Phi_{k,k-1}X_{k-1}.$ (6)

Here, $k$ signify the step of filter, $\Phi_{k,k-1}$ represents the state
transition matrix from step $k-1$ to step $k$. It can be written as
$\Phi_{k,k-1}=I+FT$, where $I$ and $F$ are identity matrix and zero matrix
respectively, $T$ is one filter step period.

The output of the accelerometer can be defined as the observation equation,

$Z_{\textrm{out,{k}}}=H_{k}{X}_{k}+V_{k},$ (7)

where $H_{k}=\tilde{A}_{k}$, and $V_{k}$ is the discrete measurement noise
that satisfies,

$E\left[V_{k}\right]=0,E\left[V_{k}V_{j}^{\textrm{T}}\right]=R_{k}\delta_{ij}.$
(8)

Here, $R_{k}$ denotes the variance matrix of the measurement noise.

The predicted observation equation is given by,

$\hat{Z}_{\textrm{out,{k}}}=H_{k}\hat{X}_{k}.$ (9)

The estimated covariance matrix is given by,

$\hat{P}_{k}=\bm{\Phi}_{k,k-1}P_{k-1}\bm{\Phi}_{k,k-1}^{\mathrm{T}}+Q_{k-1},$
(10)

where $Q$ denotes the system noise variance matrix, which is assumed to be
zero in this study.

The Kalman gain is

$K_{k}=\hat{P}_{k}H_{k}^{\mathrm{T}}\left(H_{k}\hat{P}_{k}H_{k}^{\textrm{T}}+R_{k}\right)^{\mathrm{-1}},$
(11)

and the state update value after Kalman filter is,

$X_{k}=\hat{X}_{k}+K_{k}\left(Z_{\textrm{out,{k}}}-\hat{Z}_{\textrm{out,{k}}}\right).$
(12)

Here, $\hat{X}_{k}$ represents the prior estimate value, and $X_{k}$ is the
posterior estimate.

The update of the error covariance matrix is,

$P_{k}=\left(I-K_{k}H_{k}\right)\hat{P}_{k}\left(I-K_{k}H_{k}\right)^{\mathrm{T}}+K_{k}R_{k}K_{k}^{\mathrm{T}}.$
(13)

After the Kalman filter is applied, the difference between the predicted and
observed measurements is known as the filtered residual. The filtered residual
can be used to evaluate the accuracy of the filter. The covariance matrix of
the filtered residual provides information on the uncertainty of the
estimation. Therefore, it is important to analyze both the filtered residuals
$r_{k}$ and the covariance matrix of the filtered residuals
$R_{k}^{\textrm{K}}$, as follows,

$r_{k}=Z_{\textrm{out,{k}}}-H_{k}X_{k},$ (14)
$R_{k}^{\textrm{K}}=R_{k}-H_{k}P_{k}H_{k}^{\mathrm{T}}.$ (15)

The characteristic property of the Kalman filter is that the filtered residual
vectors are uncorrelated, and even independent in the case of Gaussian
distribution.

After applying the aforementioned Kalman Filter, the state estimator of the
dynamic system can be further refined using the Rauch-Tung-Striebel (RTS)
smoother, as described in Ref. [26]. The smoothing equations are given by the
following formulas,

$\hat{X}_{k+1}=\Phi_{k+1,k}X_{k},$ (16)
$\hat{P}_{k+1}=\Phi_{k+1,k}P_{k}\Phi_{k+1,k}^{\mathrm{T}}+Q_{k},$ (17)
$G_{k}=P_{k}\Phi_{k+1,k}^{\mathrm{T}}\hat{P}_{k+1}^{\mathrm{-1}},$ (18)
$X_{k}^{\textrm{S}}=X_{k}+G_{k}\left(X_{k+1}^{\textrm{S}}-\hat{X}_{k+1}\right),$
(19)
$P_{k}^{\textrm{S}}=P_{k}+G_{k}\left(P_{k+1}^{\textrm{S}}-\hat{P}_{k+1}\right)G_{k}^{\mathrm{T}},$
(20) $r_{k}^{\textrm{S}}=Z_{\textrm{out,{k}}}-H_{k}X_{k}^{\textrm{S}},$ (21)
$R_{k}^{\textrm{S}}=R_{k}-H_{k}P_{k}^{\textrm{S}}H_{k}^{\mathrm{T}}.$ (22)

The RTS smoother provides smoothed estimates of the state mean and state
covariance at time step $k$, denoted as $X_{k}^{\textrm{S}}$ and
$P_{k}^{\textrm{S}}$, respectively. The smoother gain on time step $k$,
denoted as $G_{k}$, corrects the RTS smoother estimate. The recursion is
initialized at the last time step T of Kalman filter with
$X_{\textrm{T}}^{\textrm{S}}=X_{\textrm{T}}$ and
$P_{\textrm{T}}^{\textrm{S}}=P_{\textrm{T}}$. The smoothed residuals are
denoted as $r_{k}^{\textrm{S}}$, and the covariance matrix of smoothed
residuals is denoted as $R_{k}^{\textrm{S}}$.

To obtain a more accurate result, a combined method based on Kalman filter and
the RTS Smoother (KF-RTS) and outliers removal is proposed. The data is
filtered with the Kalman filter, and smoothed by the RTS smoother, during
which a chi-square confidence test is performed, and outliers are removed.
This KF-RTS process is iterated until there are no outliers.

A possible drawback of the filter algorithm is the fact that one needs an
initial value of the state vector together with its covariance matrix. This
can be obtained by fitting a small number of measurements at the start of the
track by a conventional least-squares fit, but this is not an elegant
solution. The other possibility is to start with an arbitrary state vector and
an infinite covariance matrix, i.e. a large multiple of the identity matrix.
This is completely in the spirit of the filtering approach, but may lead to
numerical instabilities in the computation of the gain matrix, since the
infinities have to cancel in order to give a finite gain matrix. This may be
difficult on a computer with a short word length.

In this article the initial value of the state vector is set to zero and its
covariance is chosen to be 0.001, which is large enough. The measurement noise
is calculated by selecting a segment of stationary data from the corresponding
data.

Here, we use the nonlinear least squares (NLLS) method [27] for parameter
estimation, as a crosscheck of the KF-RTS method. As is well-known, the key
challenge for NLLS is to find the value of $\hat{\theta}$ that minimizes the
function $F(\theta)$

$F(\mathbf{\theta})\equiv\frac{1}{2}\sum_{i=1}^{m}\left(f_{i}(\mathbf{\theta})\right)^{\mathrm{2}}$
(23)

where $f_{i}(\theta)\equiv y_{i}-\mathrm{Model}(\theta,\mathrm{input})$. The
Levenberg-Marquardt (LM) algorithm [28, 29], which is an efficient method to
minimize $F(\theta)$, is used to find the optimal parameters in this article.

##  V Detection and removal of outliers

During the Kalman filtering and RTS smoothing, each data point was utilized to
obtain an optimal estimate of the state value, and this process also allows
for an assessment of the quality of each data point.

The chi-square value [30] is commonly used for assessing the quality of data
and detecting outliers, which can be caused by spacecraft maneuvering, non-
Gaussian noise, electronic noise, or other coupled noise, and deviate
significantly from the normal sequence of measurements.

The residuals of the global fit can be utilized to identify measurements with
large residuals as potential outliers. Kalman filters and smoother enable the
exploitation of complete information locally, to determine the validity of a
measurement with high probability. The smoothed residual chi-square value can
serve as a useful decision criterion for data quality of the measurement,

$\chi^{2}=(r_{k}^{\textrm{S}})^{\mathrm{T}}(R_{k}^{\textrm{S}})^{\mathrm{-1}}r_{k}^{\textrm{S}}.$
(24)

It is demonstrated that the test on smoothed residual chi-square consistently
outperforms the test on filtered residual chi-square [31]. Thus, searching for
possible outliers should be carried out during smoothing, as it allows the
utilization of complete parameter information.

During the smoothing process, the measurement point $Z_{\textrm{out,{k}}}$ can
be removed from the smoothed estimate $X_{k}^{\textrm{S}}$ to obtain
${X_{k}^{\textrm{S}}}^{*}$, which represents the optimal estimate of the
system state at step $k$ using all data information except
$Z_{\textrm{out,{k}}}$. This optimal estimate can be utilized for the
detection and removal of outliers. To remove $Z_{\textrm{out,{k}}}$ from the
estimate $X_{k}^{\textrm{S}}$, an inverse Kalman filter can be applied with
the covariance matrix of $Z_{\textrm{out,{k}}}$ taken as negative. This step
of the filter is described in Ref. [31], and the smoothed estimate of $X_{k}$
without using $Z_{\textrm{out,{k}}}$, ${X_{k}^{\textrm{S}}}^{*}$, can be
calculated as,

${X_{k}^{\textrm{S}}}^{*}={X}_{k}^{\textrm{S}}+{K_{k}^{\textrm{S}}}^{*}\left(A_{\textrm{out,{k}}}-{H}_{k}{X}_{k}^{\textrm{S}}\right),$
(25)

in which

${K_{k}^{\textrm{S}}}^{*}={P}_{k}^{\textrm{S}}{H_{k}}^{\mathrm{T}}\left({H}_{k}{P}_{k}^{\textrm{S}}{H_{k}}^{\mathrm{T}}-{R}_{k}\right)^{\mathrm{-1}},$
(26)

and

${P_{k}^{\textrm{S}}}^{*}=\left({I}-{K_{k}^{\textrm{S}}}^{*}{H}_{k}\right){P}_{k}^{\textrm{S}}.$
(27)

If $Z_{\textrm{out,{k}}}$ is a valid measurement and the covariance matrix of
its Gaussian readout error is known, the quantity $\chi^{2}$ follows a chi-
square distribution with $N_{\textrm{z}}$ degrees of freedom, where
$N_{\textrm{z}}$ is the dimension of $Z_{\textrm{out,{k}}}$.

The measurement can be identified as an outlier if the value of $\chi^{2}$
exceeds a certain threshold $c$. This threshold is chosen as the $(1-\alpha)$
quantile of the corresponding $\chi^{2}$ distribution, where $\alpha$
represents the probability of rejecting a valid measurement and is chosen to
be $\alpha=0.001$ in this paper.

The measurement $Z_{\textrm{out,{k}}}$ can be removed permanently from the
list as an outlier, and the RTS smoother can continue with
${X_{k}^{\textrm{S}}}^{*}$ and ${P_{k}^{\textrm{S}}}^{*}$ instead of
${X_{k}^{\textrm{S}}}$ and ${X_{k}^{\textrm{S}}}$ for updating the estimates
$X_{j}^{\textrm{S}}$ when $j>k$. To remove all outliers, this Kalman filter
and RTS smoother must be recomputed without the outliers and iterated until
convergence is achieved.

##  VI Results of COM calibration

The COM bias estimation in this paper is performed using the two algorithms
discussed earlier. Figure 2 presents a comparison between the linear
acceleration results obtained from the NLLS estimation and the original data,
while Fig. 3 illustrates the comparison between the linear acceleration
results obtained from the extended Kalman filter algorithm and the original
data. Both methods exhibit excellent agreement with the original data. The
results of COM calibration using the KF-RTS algorithm before removing outliers
are shown in Fig. 4, with the shaded area indicating the range of one standard
deviation.

Figure 5 illustrates the comparison between the fit results obtained using the
NLLS algorithm and the experimental data with outliers removed. The final
round results of the offset obtained using the KF-RTS algorithm are presented
in Fig. 6, with the shaded area indicating the range of one standard
deviation. Figure 7 displays the comparison between the final round results
obtained using the KF-RTS algorithm and the experimental data.

Figure 2: The calibration experiment data and the NLLS fit results (with
outliers). Figure 3: The calibration experiment data, together with the first
round of outliers detected through the $\chi^{2}$ test and KF-RTS smoother
results obtained before removing the outliers. Figure 4: The first KF-RTS
smoother result of $d_{\textrm{x}}$ $d_{\textrm{y}}$ $d_{\textrm{z}}$. Figure
5: The calibration experiment data and the NLLS fit results obtained after
removing outliers. Figure 6: The final KF-RTS smoother results of
$d_{\textrm{x}}$ $d_{\textrm{y}}$ $d_{\textrm{z}}$. Figure 7: The final round
outliers detected through $\chi^{2}$ test and KF-RTS smoother result and
calibration experiment data.

Table 1 presents the estimation results for the COM offset obtained using the
two methods. Despite the use of different estimation algorithms, the results
obtained from both methods are highly consistent, thus increasing the
confidence in the accuracy of the results. It is worth mentioning that, the
value of $\chi^{2}/\mathrm{nof}$ as the goodness of a fit is 2.33 for the
first round and is 0.88 for the final round, where $\mathrm{nof}$ represents
the number of degrees of freedom.

Table 1: The results of estimation of the offset of COM. Axis | X ($\upmu$m) | Y ($\upmu$m) | Z ($\upmu$m)
---|---|---|---
NLLS with outliers |  $-$175 $\pm$ 41 | 610 $\pm$ 101 |  $-$777 $\pm$ 31
First KF-RTS |  $-$173 $\pm$ 27 | 606 $\pm$ 66 |  $-$777 $\pm$ 20
NLLS without outliers |  $-$191 $\pm$ 27 | 643 $\pm$ 66 |  $-$818 $\pm$ 20
Final KF-RTS |  $-$189 $\pm$ 27 | 638 $\pm$ 68 |  $-$818 $\pm$ 20

Figure 8 and Figure 9 depict the comparison of the amplitude spectral density
(ASD) between the calibration experimental data and flight data before and
after the COM calibration. Figure 8 shows that the peak observed in the
experimental data is caused by a modulation signal, and its influence is
highly suppressed after the COM calibration, which is consistent with our
expectations. Figure 9 demonstrates a significant reduction in the
acceleration noise level of GRS readings in the frequency range of 0.001–0.1
Hz after the COM calibration.

Figure 8: The ASD of the calibration experiment data before and after the COM
calibration. Figure 9: The ASD of the flight data before and after the COM
calibration.

##  VII Discussion and Conclusions

Detecting and reducing the deviation between the COM of the inspection load
and the COM of the satellite is crucial for high-precision accelerometers, as
it can improve their accuracy. It is also a significant step in the
development of space-based gravitational wave observatory to achieve their
scientific objectives. Furthermore, the calibration can help obtain a high-
precision gravity field, which is valuable for conducting geoscience research
with greater accuracy.

In this study, the offset of the COM between the inspection load and the
satellite is estimated using the KF-RTS smoother. Outliers are detected using
the chi-square test, and the inverse Kalman filter is applied to remove them.
The LM algorithm, as a cross check, is used to find the optimal offset
parameters for the NLLS method. The results obtained with both methods are in
very good agreement, and the offset of COM between the inspection load and the
satellite is estimated with an accuracy of $\mathcal{O}(10\,\upmu\mathrm{m})$.
After obtaining the COM offset, one can reduce it using the COM adjustment
mechanism, and the effects of the COM offset can also be suppressed in the
data-processing. The COM calibration is crucial for improving the accuracy of
the accelerometer which will directly impact the detection sensitivity of the
final space-based gravitational wave observatory.

For the space-borne gravitational-wave observatories, such as Taiji-3, there
are three satellites and each satellite is equipped with two TMs, which means
that the COM of TMs does not coincide with that of the satellite. However, as
a reference body for the satellite, it would cause residual acceleration if
the TMs are away from their nominal positions. Therefore, it is necessary to
periodically estimate or monitor the deviation of the COM of TM from a fixed
point and make adjustments when necessary. The same calibration principle and
methods as discussed in this paper can be used. The Taiji-3 satellite will be
equipped with higher-precision star-sensitive instruments, which will enable
more accurate results.

## Acknowledgements

This work is partially supported by the Strategic Priority Research Program of
the Chinese Academy of Sciences Grant No. XDA15020700 and XDA15021100, and by
the Fundamental Research Funds for the Central Universities. We acknowledge
support from the National Space Science Data Center, National Science and
Technology Infrastructure of China.

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