## 4 Results and Discussion
### Two-Stage Decoupling Simulation  
In previous experiments, we successfully detected the reference tilt using the inverted pendulum and achieved centroid calibration and reference leveling through piezoelectric ceramic-driven modulation. To accurately analyze microgravity coupled in the tilted torsion pendulum, it is essential to monitor and control changes in both roll and pitch degrees of freedom. This requires designing a two-dimensional benchmark stable platform, which presents challenges for the structure's design and assembly, as well as conditions for the decoupling control of bivariable feedback. A cascaded double pendulum is proposed to measure the thrust produced by the thruster and exclude the self-weight superposed in the inclined torsional pendulum. As shown in Figure 16, the first torsion pendulum TP1, fixed on the installation plane, serves as the front stage pendulum. The relative position between TP1 and the table is maintained constant through null position control. The second torsion pendulum, TP2, also known as the rear torsion pendulum, is fixed at one end of the front pendulum, and detects the change in thrust to be measured in displacement mode. When the base plane is tilted, the micro-thrust torque of the front and rear pendulums is the same, but the microgravity coupling of the two pendulums differs due to the tilt and eccentricity. The self-weight of TP1 is greater than that of TP2, and the center of mass deviation of TP1 is also larger than that of TP2. This can enhance the combined microgravity and serve as the reference tilt of the amplifier mechanism. The combined stiffness of two flexural hinges connected in series can be calculated as the stiffness of TP1, or \(K_{1}\), in parallel with the stiffness of TP2, or \(K_{2}\), i.e.,  
\[\frac{1}{K_{c}}=\frac{1}{K_{1}}+\frac{1}{K_{2}} \tag{13}\]  
In design, the elastic stiffness TP1 and TP2 can be selected as the same nominal stiffness \(K_{0}\), so that \(K_{c}=K_{0}/2\). However, the motion results for the double pendulum under both open-loop stages tend to be chaotic. Therefore, TP1 is adjusted by null type control, allowing \(K_{1}>>K_{2}\) to be maintained in the low-frequency range. Thus, the ultimate composite stiffness is \(K_{c}=K_{0}\). Figure 16: The concept model of a two-stage torsional pendulum. For a semi-open-loop and semi-close-loop double pendulum, both have an equivalent torque. The former is an electromagnetic balanced force equal to the micro thrust, and the latter is an elastic restoring force equal to the micro thrust under nominal stiffness. In other words, they share a common component \(\tau_{0}=FL\). After tilting, TP1 experiences a higher gravity torque due to the larger centroid displacement and greater self-weight, combined with the tilt of the common base, represented by:  
\[\tau_{1}=(M_{1}+M_{2})g\sin(\alpha)L_{cm_{1}} \tag{14}\]  
TP2 experiences a slight gravitational coupling torque due to the same tilt of the common base but has a much smaller displacement of the centroid, as indicated by the following:  
\[\tau_{2}=M_{2}g\sin(\alpha)L_{cm_{2}} \tag{15}\]  
Therefore, TP1 will be more sensitive to the reference frame's tilt compared to TP2 if  
\[\frac{\Delta\tau}{\tau_{2}}=\frac{\tau_{1}-\tau_{2}}{\tau_{2}}\approx\frac{( M_{1}+M_{2})L_{cm_{1}}}{M_{2}L_{cm_{2}}}>>1 \tag{16}\]  
We could simulate this in Simscape Multibody to verify. As depicted in Figure 17, the two-stage pendulum model exhibits a basic tilt, with the platform being modulated at a frequency of 0.01 Hz and an amplitude of 3 urad. The differential and extracted force signal used for balancing the tilt bias of TP1 can also be used as a reference for decoupling the microgravity on TP2, thereby recovering a square wave that closely matches the test input thrust. It is evident that utilizing a two-stage pendulum for semi-closed-loop and semi-open-loop measurements can effectively integrate the coupling of pitch and roll directions of gravity, enhance measurement accuracy and stability, and reduce platform thrust noise.