## 2 Device and Method
### Modeling of Gravity Coupling  
According to Equation (3), the torque stiffness of the torsional pendulum under the action of thrust \(F\) can be obtained as:  
\[K=\frac{FL}{\theta}=K_{0}-\frac{Mgsin(\alpha)l_{cm}}{\theta}, \tag{4}\]  
where, the nominal stiffness \(K\) includes the flexural pivot's elastic stiffness \(K_{0}\) and the gravity stiffness \(\frac{Mgsin(\alpha)l_{cm}}{\theta}\). The former is determined by the pivot material and is independent of the applied force within a certain range. The latter is not only connected to the mass of the pendulum frame, the gravitational acceleration, the reference inclination, and the centroid offset, but also to the deflection angle \(\theta\) of the torsional pendulum caused by the measured thrust. However, analyzing the variation of stiffness with the inclination of the mounting reference is not straightforward due to the fact that the inclination \(\alpha\) also affects the angular deformation of the torsional pendulum.  
The compliance is the reciprocal of stiffness and represents the (angular) displacement generated by a unit force (torque) in the system [18]. In the previous discussion, for a  
Figure 1: Definition of the coordinate system of torsion pendulum (isometric view).  
Figure 2: Angle definition of torsion pendulum roll direction (side view).  
specific thrust \(F\), under the combined action of pivot elasticity and pendulum frame gravity, the torque compliance of the torsional pendulum system can be expressed as:  
\[S=\frac{1}{K_{0}}+\frac{Mgssin(\alpha)l_{cm}}{FL}. \tag{5}\]  
In this equation, there are no other variables related to \(\alpha\). Under the small angle approximation, the linear relationship between compliance and inclination becomes simpler. During this process, the stiffness of the torsional pendulum varies according to \(K=\frac{1}{5}\), and Equation (4) can be rewritten as:  
\[K=\frac{1}{1+\frac{Mgssin(\alpha)l_{cm}}{FLK_{0}}}. \tag{6}\]  
The physical interpretation of this equation is more intricate. Ideally, \(K=K_{0}\) when \(Mgssin(\alpha)l_{cm}=0\). However, we need to consider at least three cases when \(Mgssin(\alpha)l_{cm}\
eq 0\).  
1. \(\frac{Mgssin(\alpha)l_{cm}}{FL}>0\): In this case, \(0<K<K_{0}\) and the torque from the thrust and gravity act in the same direction. For example, when a force is applied to the torsional pendulum, although it experiences a restoring force from the pivot, the stiffness gradually decreases as the gravitational component introduced by the inclination increases.
2. \(-1<\frac{Mgssin(\alpha)l_{cm}}{FL}<0\): In this case, the torque from the thrust and gravity act in opposite directions, but their combined torque is in the same direction as the thrust torque. The elastic torque and the gravity torque also act in the same direction, which means that the torsional pendulum and the gravity pendulum are superimposed. In this scenario, \(K>K_{0}\). Particularly, when \(\frac{Mgssin(\alpha)l_{cm}}{FL}\rightarrow-1\), the stiffness increases sharply.
3. \(\frac{Mgssin(\alpha)l_{cm}}{FL}<-1\): In this case, the torque from the thrust and gravity act in opposite directions, but their combined torque is in the same direction as the gravity torque. The elastic torque and the thrust torque also act in the same direction, resulting in a reverse stiffness of the system, \(K<0\). This is beyond the scope of this discussion.  
It can be seen that under the condition of reference tilt, with the influence of centroid deviation, the stiffness change law of the torsion pendulum is not intuitive. Therefore, this paper will first use a Simscape visual modeling tool for semi-physical simulation and error analysis, to fit the numerical effects of the reference tilt and centroid shift. In particular, by simulating the thrust and displacement time series of the torsion pendulum in the process of gravity coupling, the thrust-displacement transfer function of the system is identified, and the variation law of its compliance and displacement with the center of gravity and tilt is obtained.