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Calibration of key parameters during the in-orbit phase for the Taiji-2 gravitational reference sensor

This paper presents an advanced in-orbit calibration framework for the Taiji-2 gravitational reference sensor that utilizes spacecraft maneuvers and Kalman filtering to simultaneously estimate scale factors and center-of-mass offsets with high precision, thereby ensuring the mission's sensitivity requirements and establishing a scalable paradigm for future gravitational wave observatories.

Original authors: Haoyue Zhang, Chang Liu, Xiaotong Wei, Peng Xu, Li-E Qiang, Ziren Luo, Ye Dong

Published 2026-03-27

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CC BY 4.0

Original authors: Haoyue Zhang, Chang Liu, Xiaotong Wei, Peng Xu, Li-E Qiang, Ziren Luo, Ye Dong

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a whisper in a hurricane. That is essentially what the Taiji mission is trying to do, but instead of a hurricane, it's the chaotic noise of space, and instead of a whisper, it's a gravitational wave—a tiny ripple in the fabric of the universe caused by massive events like black holes colliding.

To hear this "whisper," the Taiji satellite needs to be incredibly sensitive. It carries a special sensor called a Gravitational Reference Sensor (GRS). Inside this sensor floats a heavy, gold-coated cube (the "Test Mass") that is completely free-falling, untouched by anything except gravity. The satellite itself acts like a protective shell, using tiny thrusters to hover around this floating cube without ever touching it.

However, there's a problem. Over time, the tools we use to measure things get slightly "drunk" or confused.

The Two Main Problems

The paper tackles two specific ways the satellite's "eyes" and "balance" can get messed up:

The "Ruler" is Stretching (Scale Factors):
Imagine you have a ruler to measure a table. If the ruler suddenly stretches by 1%, your measurement will be wrong, even if you are looking perfectly straight. In the satellite, the "ruler" is the scale factor. It's the mathematical conversion that turns an electrical voltage reading into a real-world distance or acceleration. If this conversion factor drifts due to heat or aging electronics, the satellite thinks it's moving when it's actually still, or vice versa.
The "Center of Balance" is Shifting (Center-of-Mass Offsets):
Imagine a spinning top. If its weight is perfectly centered, it spins smoothly. But if you add a tiny pebble to one side, it wobbles. The satellite is the top, and the floating cube is the weight. If the floating cube isn't perfectly centered relative to the satellite's main body, the satellite's own movements (like turning its head to look at stars) will create "fake" vibrations that look like gravitational waves. This is the Center-of-Mass (c.m.) offset.

The Solution: The "Dance" and the "Calculator"

The authors of this paper developed a clever way to fix these problems while the satellite is already floating in space. They didn't just guess; they created a calibration dance.

Step 1: The Wiggle (The Maneuver)
The satellite uses its tiny thrusters to perform a specific, rhythmic "wiggle." It swings back and forth in a square-wave pattern (like a metronome ticking: tick-tock, tick-tock). This is a controlled, known movement.

Step 2: The Two Witnesses (Star Tracker & GRS)
As the satellite wiggles, two things happen simultaneously:

The Star Tracker (The Eye): This is like a high-tech camera looking at the stars. It sees exactly how much the satellite is turning and how fast. It knows the "truth" of the movement.
The GRS (The Ear): This sensor feels the forces. It measures the voltage caused by the floating cube reacting to the wiggle.

Step 3: The Detective Work (Kalman Filter)
Here comes the magic. The scientists use a mathematical tool called a Kalman Filter. Think of this as a super-smart detective who listens to both witnesses.

The detective compares what the "Eye" (Star Tracker) says the satellite did with what the "Ear" (GRS) felt.
If the "Ear" says, "I felt a huge shake," but the "Eye" says, "We only turned a tiny bit," the detective knows the "Ruler" (Scale Factor) is broken.
If the "Ear" feels a weird wobble that doesn't match the "Eye's" smooth turn, the detective knows the "Center of Balance" (c.m. offset) is off.

By running this dance for a few hours, the detective can calculate exactly how much the ruler has stretched and exactly how far the balance point has shifted.

The Results: A Perfect Tune-Up

The paper simulates this process for two different designs of the Taiji-2 satellite. The results are impressive:

The Ruler: They corrected the scale factors with an error of less than 0.2%. This is like measuring the length of a football field and being off by less than the width of a human hair.
The Balance: They located the center of mass with an error of less than 100 micrometers (about the width of a human hair).

Why This Matters

This isn't just about fixing one satellite. It's about proving that we can keep our "cosmic ears" tuned perfectly while floating in space for years.

For Taiji-2: This ensures the mission can actually hear the gravitational waves it was built to find.
For the Future (Taiji-3): It sets a blueprint for even more sensitive missions. If we can calibrate the sensors this well now, the next generation of satellites can be even more precise, potentially letting us hear the "whispers" of the very beginning of the universe.

In short, the authors figured out how to teach a satellite to "check its own homework" while it's in space, ensuring that when it finally listens to the universe, it's listening with perfect clarity.

1. Problem Statement

The Taiji mission is a Chinese space-based gravitational wave observatory designed to detect signals in the 0.1 mHz to 1 Hz frequency range. To achieve its scientific goals, the mission requires an ultra-precise Gravitational Reference Sensor (GRS) with a residual acceleration noise level below $3 \times 10^{-14} \, \text{m s}^{-2} \text{Hz}^{-1/2}$ (for Taiji-2).

Two critical parameters degrade the performance of the GRS over time due to propellant consumption, thermal effects, and electronic aging:

Scale Factors: The conversion coefficients that translate raw voltage readouts from the sensor electrodes into physical linear and angular accelerations.
Center-of-Mass (c.m.) Offsets: The physical displacement between the Test Mass (TM) and the Spacecraft (SC) center of mass.

If these parameters are not calibrated with high precision in orbit, the coupling between the spacecraft's attitude motion (jitter) and the c.m. offset introduces acceleration noise that exceeds the mission's sensitivity requirements. Specifically, the scale factor error must be $<0.3\%$ , and the c.m. offset calibration error must be within 100 µm.

2. Methodology

The authors propose an advanced in-orbit calibration framework that simultaneously estimates both the scale factors and c.m. offsets using a combination of controlled spacecraft maneuvers and a Kalman Filter.

A. Operational Principle

Maneuver Strategy: The spacecraft executes controlled attitude swings by applying periodic square-wave torque signals via micro-thrusters.
Data Sources:
- Star Tracker (ST): Provides high-precision measurements of the spacecraft's angular velocity ( $\omega_{SC}$ ) and angular acceleration ( $\dot{\omega}_{SC}$ ).
- GRS Readouts: Provides voltage signals ( $V$ ) from the electrode cages.
Decoupling Mechanism:
1. Scale Factor Calibration: By correlating the known angular acceleration from the ST with the GRS voltage readouts, the angular scale factors ( $G_{angle}$ ) are estimated.
2. c.m. Offset Calibration: Using the calibrated scale factors, the GRS voltage is converted to linear acceleration. This reconstructed acceleration is then compared against the ST-derived angular motion to solve for the c.m. offset vector ( $\vec{d}$ ). The relationship is governed by the equation: $\vec{a}_{coupling} \approx \dot{\vec{\omega}} \times \vec{d}$ .

B. Simulation Setup

Spacecraft Configurations: The study simulates two potential Taiji-2 layouts:
- Type-1: SC c.m. located at the midpoint between the two TMs.
- Type-2: SC c.m. located at the intersection of the two laser links.
Noise Modeling: The simulation incorporates realistic noise sources, including ST readout noise, thruster noise, Low-Frequency Suspension (LFS) control noise, and solar radiation pressure (SRP) fluctuations.
Algorithm: A discrete Kalman Filter is employed to estimate the state vectors (scale factors and c.m. offsets) recursively, minimizing the impact of measurement noise.

3. Key Contributions

Simultaneous Estimation: The paper presents the first scheme tailored for Taiji-2 that simultaneously calibrates scale factors and c.m. offsets in a single experiment, optimizing propellant usage.
Dual-Parameter Framework: It establishes a robust mathematical model that decouples the coupled disturbances (attitude motion and c.m. offset) using the distinct signatures of angular acceleration and linear acceleration.
Configuration Robustness: The methodology is validated against two distinct spacecraft geometric configurations (Type-1 and Type-2), demonstrating that the calibration logic holds regardless of the specific c.m. placement.
Noise Budget Analysis: The study rigorously defines the noise budget (payload, platform, and environmental) required to achieve the sub-micrometer calibration precision.

4. Results

The numerical simulations yielded the following results for both Type-1 and Type-2 spacecraft configurations:

Scale Factor Accuracy:
- The estimation error for angular scale factors ( $G_{\phi}, G_{\theta}, G_{\psi}$ ) was consistently below 0.2%.
- This satisfies the Taiji-2 requirement of $\le 0.3\%$ .
- The linear scale factors, derived from the angular ones, also met the mission requirements.
c.m. Offset Accuracy:
- The residual errors for the c.m. offsets along all three axes ( $X, Y, Z$ ) were within 100 µm.
- For example, in Type-1, the estimated error for the X-axis was $121.0 \pm 70.3$ µm (mean error within bounds), and similar precision was achieved for Y and Z axes.
- This meets the strict requirement of $\le 100$ µm.
Convergence: The Kalman filter demonstrated rapid convergence, with the estimated values stabilizing within the required error bounds over the simulated maneuver period (250-second square wave cycles).

5. Significance

Mission Feasibility: The results confirm that the Taiji-2 mission can achieve its target sensitivity of $3 \times 10^{-14} \, \text{m s}^{-2} \text{Hz}^{-1/2}$ by effectively mitigating acceleration noise caused by c.m. offsets and scale factor drifts.
Scalability for Future Missions: The proposed calibration paradigm is directly applicable to the more advanced Taiji-3 mission, which will require even stricter stability (sub-micrometer c.m. stability and ultra-low noise).
Technological Advancement: This work bridges the gap between theoretical requirements and practical in-orbit operations, providing a proven methodology for maintaining the ultra-high precision necessary for space-based gravitational wave astronomy.

In conclusion, the paper validates a robust, propellant-efficient, and high-precision calibration strategy essential for the success of the Taiji gravitational wave observatory.