## 3 Numerical Analysis  
### Multibody Simulation  
In Simscape Multibody, a multibody system of a torsional pendulum is modeled using modules representing objects, joints, constraints, force elements, and sensors, making it possible to build a dynamic model of the entire structure and solve its equations of motion. Based on the finite element modeling method, the reference inclination, centroid offset, and other parameter attributes are changed by MATLAB scanning, different thrust is excited in the model, and the simulation model of the given parameters is run to collect the discrete thrust and displacement time series data. Then, the system identification method, such as the least square method, is used to obtain the thrust--displacement transfer function, and the value of its transfer function at the frequency of 0 is viewed. The study was conducted on different stiffness values of 0.2 Nm/rad and 0.02 Nm/rad, and the results are shown in Figures 3 and 4. First, regardless of the gravity coupling and resolution force magnitude, the eigenfrequency peaks corresponding to different stiffness values in the amplitude spectrum of the thrust-displacement transfer function are different. Under the condition of constant mass or moment of inertia of the pendulum frame, the low-stiffness torsional pendulum has a higher overall compliance level near 0 Hz. Second, for the case of a smaller resolution force of \(F=0.1\) uN, the gravity coupling caused by the same angle of inclination is more uncertain, resulting in a more discrete compliance obtained through position calibration. This is more pronounced in the case of low stiffness. From the perspective of the compliance variation caused by the inclination of the pendulum frame, lower resolution force and lower stiffness both contribute to thrust noise, and determining the threshold of reference inclination required for different resolution force levels becomes more critical. To this end, simulation data of a torsional pendulum with a nominal stiffness of 0.2 \(\,\mathrm{Nm}/\mathrm{rad}\) and a resolution force of 1 uN square wave thrust, with the same center of mass offset, were used to fit the stiffness and compliance values for different pitch angles ranging from \(-20\) urad to \(20\) urad. These angles are chosen to cover the platform's tilt range during thermal expansion and contraction over a long period of time. For instance, using an iron quadruped platform as support, when the temperature fluctuates within \(\pm 2\)\({}^{\circ}\)C, which is within the predictable range, the material's thermal expansion coefficient is \(12.2\times 10^{-6}/\mathrm{K}\). When the fluctuation on a 1-meter-long plate exceeds \(\pm 25\)\(\,\mathrm{\SIUnitSymbolMicro m}\), the inclination angle approaches \(\pm 20\)\(\,\mathrm{\SIUnitSymbolMicro rad}\). Moreover, depending on whether the torsion pendulum remains stable as gravitational coupling increases, there will be convergence due to changes in stiffness or flexibility within this range. Despite an obvious divergence of \(\pm 20\)\(\,\mathrm{\SIUnitSymbolMicro rad}\) mentioned in the paper and considering a resolution of 0.1 uN, overall calculations remain valid. Figure 3: Transfer function from thrust to displacement with stiffness values of 0.2 \(\,\mathrm{Nm}/\mathrm{rad}\) and 0.02 \(\,\mathrm{Nm}/\mathrm{rad}\) at thrust 1 \(\,\mathrm{\SIUnitSymbolMicro N}\). As shown in Figure 5, the stiffness varies with the reference inclination in a binomial exponential fitting relationship, while as shown in Figure 6, the compliance has a linear relationship with the corresponding inclination. It can be seen that the stiffness variation curve caused by inclination is neither symmetric nor linear, and the binomial exponential form represents the process of the elastic stiffness of the torsional pendulum evolving into gravitational stiffness. For a specific resolution force, the upper and lower limits of the corresponding pitch angles can be inferred based on the stiffness variation represented by interpolated exponential terms.