## 3 Numerical Analysis
### Tilt Angle Monitoring  
In the process of measuring micro thrust using a torsional pendulum, it is sensitive to the inclination of the platform reference. The sensitivity of the torque of the measured thrust to the platform tilt angle is on the order of 0.01 \(\upmu\)N - m/ urad or even lower. Conventional commercial tilt sensors have limited detection accuracy. For example, MEMS sensors can only resolve the smallest tilt angle of 0.001\({}^{\circ}\) (approximately 17 \(\upmu\)rad), which is insufficient for detecting and compensating for small tilts. Therefore, it is necessary to design a mechanism that amplifies the reference tilt angle, which can not only resolve the weak changes in thrust but also provide a good reference for the platform tilt and the direction of the gravitational field. Figure 7 shows a reference tilt amplification mechanism designed based on the inverted pendulum principle [19; 20]. Figure 4: Transfer function from thrust to displacement with stiffness values of 0.2  Nm/rad and 0.02  Nm/rad at thrust 0.1 \(\upmu\)N.  
For the inclined inverted pendulum, considering only the balance between gravity and elasticity, the equation can be written as:  
\[mgL_{cm}\sin\theta+\varphi=k\theta. \tag{7}\]  
Figure 5: Thrust stiffness fitting at different pitch angles. Figure 6: Thrust compliance fitting at different pitch angles. where \(m\) is the mass of the inverted pendulum, \(L_{cm}\) is the length of the center of mass position arm, \(\theta\) is the deviation angle of the inverted pendulum, \(\varphi\) is the inclination angle of the reference surface, and \(k\) is the torque stiffness coefficient. This equation describes the balanced relationship between the gravity, deviation angle, and reference tilt angle of the inverted pendulum. In the small angle approximation, \(\sin(\theta+\varphi)\approx\theta+\varphi\), and the balance equation can be simplified as:  
\[mgL_{cm}\theta+\varphi\approx k\theta. \tag{8}\]  
According to the above equation, the ratio between the deformation angle \(\theta\) and the reference tilt angle \(\varphi\) of the inverted pendulum can be expressed as:  
\[N=\frac{\theta}{\varphi}=\frac{1}{k-mgL_{cm}}. \tag{9}\]  
When the gravity stiffness \(mgL_{cm}\) of the inverted pendulum is close to its elastic deformation stiffness \(k\), \(N\) will rapidly increase. This means that the relationship between the deviation angle of the inverted pendulum and the normal tilt angle of the reference surface becomes more significant. This amplifies the detection of small tilt angles, but if the ratio \(N\) becomes too large, it will lead to a rapid increase in the structural instability of the inverted pendulum. In practical applications, the ratio \(N\) is usually controlled within a reasonable range, so as to amplify small tilt angles while maintaining the stability of the structure. Therefore, it is necessary to choose appropriate gravity stiffness and elastic deformation stiffness based on specific application requirements and structural parameters, in order to balance the relationship between amplification detection and structural stability. In theory, the gravity stiffness of a pendulum depends on the weight of the mass center (\(mg\)) and its distance from the axis (\(L_{cm}\)), while the elastic stiffness of a flexural pivot is determined by its dimensions and material. However, in practice, these values are simulated by designers or calibrated through experiments, such as adjusting the counterweight to modify the period of the inverted pendulum, in order to achieve the desired reference tilt amplification ratio. For example, the amplification factor \(N\) is approximately 10 when \(k=0.1\,\mathrm{Nm}/\mathrm{rad}\) and \(mgL_{cm}=0.09\,\mathrm{Nm}/\mathrm{rad}\). It is worth noting that, after the weight increases to a certain  
Figure 7: Tilt amplifying mechanism based on an inverted pendulum.