## 5 Conclusions  
The torsion pendulum is an important tool for ground testing of micro-thrusters. However, calibrating low-frequency micro-Newton level thrusts below 10 mHz is challenging due to load heating, material strain, and ground vibrations. The background noise from the pendulum frame is difficult to overcome, and the microgravity coupling caused by baseline tilt is a significant factor contributing to the time-varying stiffness. This study systematically analyzes the dynamics of the torsion pendulum under platform tilt conditions. It conducts multibody simulations and numerical analyses on the equivalent center of mass offset model to determine the changes in stiffness and compliance resulting from platform tilt. The study suggests utilizing an inverted pendulum to enhance the micro angle of baseline tilt. Additionally, a leveling base composed of piezoelectric actuator and a lifting platform is designed. The feasibility of detecting, modulating, and controlling the tilt of the pendulum frame is experimentally demonstrated. The main innovations of this study include:  
1. The dynamic equations of the torsion pendulum are reconstructed by adding a microgravity additional term that couples tilt and eccentricity to the static equilibrium. The stiffness of the deformation under test is decomposed into elastic stiffness and gravity stiffness, and the influence of center of mass offset and baseline tilt is linearized using the concept of compliance. 2. Using Simscape Multibody creatively to simulate the coupling effect of microgravity in the torsion pendulum, identifying the thrust position transfer function of the system, fitting the stiffness variation with a binomial exponential, and determining the error threshold of various factors that match the resolution force. 3. Using an inverted pendulum to detect the baseline tilt of the torsion pendulum, studying the optimization of the relationship between the counterweight and deformation and tilt amplification, and numerically analyzing the center of mass calibration based on amplitude-modulated thrust-tilt. 4. For the roll coupling of the torsion pendulum microgravity, the design of a baseline leveling mechanism driven by piezoelectric actuator is proposed. The effectiveness of detecting, modulating, and controlling the tilt change of the base is demonstrated through experiments without drag control. 5. The platform's 2D tilt instability and the risk of baseline leveling failure can be reduced by using a two-stage pendulum to separate tilt and thrust. This approach has been preliminary simulated and validated, and will be further tested in future research. In conclusion, based on the research work presented in this paper, we have learned that through simulation calculations and modulation testing, the center of mass position and platform tilt can be calibrated, and the torque applied by the pendulum frame gravity can be removed from the results, reducing the micro-thrust noise from thermal strain and ground vibrations. In preliminary experiments, after drag-free control of the platform, the tilt drift of the inverted pendulum at 0.1 mHz decreased from 5.7 m/Hz\({}^{1/2}\) to 0.28 m/Hz\({}^{1/2}\), meeting the requirements for higher differential force accuracy and noise testing. In the future, we will further test the significant reduction in low-frequency noise of micro-thrusters under the expected baseline leveling and center of mass calibration under thermal load influence. Conceptualization, L.C. and C.Q. ; methodology, L.C. ; software, L.C. ; validation, L.C., J.W. and J.L. ; formal analysis, L.C. ; investigation, L.C. ; resources, C.Q. ; data curation, C.Q. ; writing--original draft preparation, L.C. ; writing--review and editing, L.C. ; visualization, H.D. ; supervision, J.W. ; project administration, J.M. ; funding acquisition, C.Q. All authors have read and agreed to the published version of the manuscript.